THREE ESSAYS ON MORE POWERFUL UNIT ROOT TESTS

WITH NON-NORMAL ERRORS

by

MING MENG

JUNSOO LEE, COMMITTEE CHAIR

ROBERT REED JUN MA

SHAWN MOBBS MIN SUN

A DISSERTATION

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy

in the Department of Economics, Finance, and Legal Studies in the Graduate School of

The University of Alabama

TUSCALOOSA, ALABAMA

2013

Copyright Ming Meng 2013 ALL RIGHTS RESERVED

ii

ABSTRACT

This dissertation is concerned with finding ways to improve the power of unit root tests.

This dissertation consists of three essays. In the first essay, we extends the Lagrange Multiplier

(LM) unit toot tests of Schmidt and Phillips (1992) to utilize information contained in non-

normal errors. The new tests adopt the Residual Augmented Least Squares (RALS) estimation

procedure of Im and Schmidt (2008). This essay complements the work of Im, Lee and Tieslau

(2012) who adopt the RALS procedure for DF-based tests. This essay provides the relevant

asymptotic distribution and the corresponding critical values of the new tests. The RALS-LM

tests show improved power over the RALS-DF tests. Moreover, the main advantage of the

RALS-LM tests lies in the invariance feature that the distribution does not depend on the

nuisance parameter in the presence of level-breaks.

The second essay tests the Prebisch-Singer hypothesis by examining paths of primary

commodity prices which are known to exhibit multiple structural breaks. In order to examine the

issue more properly, we first suggest new unit root tests that can allow for structural breaks in

both the intercept and the slope. Then, we adopt the RALS procedure to gain much improved

power when the error term follows a non-normal distribution. Since the suggested test is more

powerful and free of nuisance parameters, rejection of the null can be considered as more

accurate evidence of stationarity. We apply the new test on the recently extended Grilli and

Yang index of 24 commodity series from 1900 to 2007. The empirical findings provide

significant evidence to support that primary commodity prices are stationary with one or two

iii

trend breaks. However, compared with past studies, they provide even weaker evidence to

support the Prebisch-Singer hypothesis.

The third essay extends the Fourier Lagrange Multiplier (FLM) unit root tests of Enders

and Lee (2012a) by using the RALS estimation procedure of Im and Schmidt (2008). While the

FLM type of tests can be used to control for smooth structural breaks of an unknown functional

form, the RALS procedure can utilize additional higher-moment information contained in non-

normal errors. For these new tests, knowledge of the underlying type of non-normal distribution

of the error term or the precise functional form of the structure breaks is not required. Our

simulation results demonstrate significant power gains over the FLM tests in the presence of

non-normal errors.

iv

DEDICATION

This dissertation is dedicated to those who wholeheartedly supported and guided me,

especially to my parents and my wife.

v

ACKNOWLEDGMENTS

I would like to express the deepest appreciation to my committee chair Professor Junsoo

Lee, who continually and convincingly conveyed an enthusiasm in regard to research and

scholarship, and an excitement in regard to teaching. Without his guidance and help this study

would not have been possible.

I would like to thank my committee members, Professor Robert Reed, Professor Jun Ma,

Professor Shawn Mobbs, and Professor Min Sun, for their invaluable comments, inspiring

questions, and kind support for my dissertation. In addition, a thank you to Professor Walter

Enders, who introduced me to Time Series Econometrics. I also wish to thank Professor

Matthew Holt who helped me a lot. Their support was greatly appreciated.

Finally, and importantly, I wish to give my thanks to my wife Xinyan Yan who has

supported me and listened to my complaints.

vi

CONTENTS

ABSTRACT ............................................................................................................ ii

DEDICATION ....................................................................................................... iv

ACKNOWLEDGMENTS .......................................................................................v

LIST OF TABLES ............................................................................................... viii

LIST OF FIGURES ............................................................................................... ix

INTRODUCTION ...................................................................................................1

1. More Powerful LM Unit Root Tests with Non-normal Errors ............................4

1.1. LM and RALS-LM Tests .....................................................................6

1.2. Simulations ........................................................................................13

1.3. Concluding Remarks ...........................................................................15

2. The Prebisch-Singer Hypothesis and Relative Commodity Prices: Further Evidence on Breaks and Trend Shifts...................................................22

2.1. Econometric Methodology..................................................................25

2.2. Monte Carlo Experiment.....................................................................32

2.3. Data and Empirical Results .................................................................33

2.4. Concluding Remarks ...........................................................................38

3. More Powerful Fourier-LM Unit Root Tests with Non-Normal Errors ............48

3.1. The RALS-FLM Test .........................................................................52

3.2. The Monte Carlo Experiments ............................................................57

3.3. Concluding Remarks ..........................................................................60

vii

3.4. Appendix .............................................................................................61

CONCLUSION .....................................................................................................70

REFERENCES .....................................................................................................74

viii

LIST OF TABLES

1.1.Critical Values of RALS-LM Test with No Break ..............................17

1.2. Critical Values of RALS-LM Test with Level Break .........................18

1.3. Size, Power, and Size-Adjusted power with No Break ......................19

1.4. Size Power, and Size-Adjusted Power with Level Break ...................20

2.1. Results using ADF Test and No-Break LM Unit Root Tests .............40

2.2. Results using Transformed One-Break LM and RALS-LM Tests .....41

2.3. Results using Transformed Two-Break LM and RALS-LM Tests ....42

2.4. Results using Two-Step LM and Three-Step RALS-LM Tests ..........43

2.5. Estimated Trend Stationary and Difference Stationary Models .........44

2.6. Relative Measures of a Prevalence of a Trend ....................................45

3.1.a Critical Values of Tests .................................................63

3.1.b Critical Values of ...............................................64

3.2. Critical Values of .......................................................64

3.3. Finite Sample Performance with Known Frequencies........................65

3.4. Finite Sample Performance of Tests using Tests .......................67

3.5. Finite Sample Performance using Cumulative Frequencies ...............69

ix

LIST OF FIGURES

1.1. Size-Adjusted Power ...........................................................................21

2.1. Relative Primary Commodity Prices ..................................................46

1

INTRODUCTION

The main focus of this dissertation is to find ways to improve the power of unit root tests.

The issue of whether an economic time series is best characterized by either a unit root or a

stationary process has assumed great importance in both the theoretical and the applied time

series econometrics literature. As a consequence, tests of the null hypothesis that a series is

integrated of order one, I(1), against the alternative hypothesis that it is integrated of order zero,

I(0), have received much attention. It is well known that traditional unit root tests have poor

power, or in other words, the capability of traditional unit root tests to distinguish in finite

samples the unit root null from nearby stationary alternatives is low. As such, the development

of more powerful unit root tests has not been a trivial concern. In a seminal paper, Perron (1989)

first documented that failure to account for a structural break can result in the standard Dickey-

Fuller (DF) test lose power significantly. To provide a remedy, Perron suggested a modified DF

unit root test that includes dummy variables to control for known structural breaks in the level

and slopes. Perron’s study has three assumptions that are hardly satisfied in empirical

researches, which includes: (i) the structural breaks are assumed to be known a priori, (ii) the

structural breaks are assumed to be steep or abrupt, and (iii) the error term is assumed to follow a

standard normal distribution, respectively. Subsequent papers further modified unit root tests

have been developed to relax one or more of these assumptions. See Zivot and Andrews (1992),

Lumsdaine and Papell (1997), Perron (1997), Vogelsang and Perron (1998), and Lee and

Strazicich (2003, 2004), among others.

In the first essay, we extends the Lagrange Multiplier (LM) unit toot tests of Schmidt and

2

Phillips (1992) to utilize information contained in non-normal errors. The new tests adopt the

Residual Augmented Least Squares (RALS) estimation procedure of Im and Schmidt (2008).

This essay complements the work of Im, Lee and Tieslau (2012) who adopt the RALS procedure

for DF-based tests. We refer them as the RALS-LM tests. This essay provides the relevant

asymptotic distribution and the corresponding critical values of the new tests. The RALS-LM

tests show improved power over the RALS-DF tests, and the power of both RALS-DF and

RALS-LM tests can increase dramatically when the error term is highly asymmetric or has fat-

tails with unknown forms of non-normal distributions. Moreover, the main advantage of the

RALS-LM tests lies in the invariance feature that the distribution does not depend on the

nuisance parameter in the presence of multiple level-breaks, and it is expected that they can be

more useful in extended models with other types of structural changes.

The second essay tests the Prebisch-Singer hypothesis by re-examining paths of primary

commodity prices which are known to exhibit multiple structural breaks. In order to examine the

issue more properly, we first suggest new unit root tests that can allow for structural breaks in

both the intercept and the slope. In particular, we adopt a procedure to make the resulting test

free of the nuisance parameter problem which trend-shifts induce otherwise. Then, we adopt the

RALS procedure to gain much improved power when the error term follows a non-normal

distribution. Since the suggested test is more powerful and free of nuisance parameters, rejection

of the null can be considered as more accurate evidence of stationarity. We apply the new test

on the recently extended Grilli and Yang index of 24 commodity series from 1900 to 2007. The

main findings of this study reveal that 21 out of the 24 commodity prices are found to be

stationary around a broken trend, implying that shocks to these commodities tend to be

transitory. Only three relative commodity price series are found to be difference stationary. The

3

empirical findings provide significant evidence to support that primary commodity prices are

stationary with one or two trend breaks. In our trend analysis, we find that only 7 series in which

the relative commodity prices display negative trend more than 50% of the time period

examined; however, 8 relative commodity prices display no significant positive or negative trend

in more than 90% of the time period examined. Compared with past studies, our findings

provide even weaker evidence to support the Prebisch-Singer hypothesis.

The third essay extends the Fourier Lagrange Multiplier (FLM) unit root tests of Enders

and Lee (2012a) by using the RALS estimation procedure of Im and Schmidt (2008). While the

FLM type of tests can be used to control for smooth structural breaks of an unknown functional

form, the RALS procedure can utilize additional higher-moment information contained in non-

normal errors. For these new tests, knowledge of the underlying type of non-normal distribution

of the error term or the precise functional form of the structure breaks is not required. By using

parsimonious number of parameters to control for possible breaks in the deterministic term and

use the additional information lied in nonlinear moments of the error term, the power of RALS-

F-LM test increase dramatically when the error term follows a non-normal distribution.

4

CHAPTER 1

MORE POWERFUL LM UNIT ROOT TESTS WITH NON-NORMAL ERRORS

A recent paper of Im, Lee and Tieslau (2012) adopts the Residual Augmented Least

Squares (RALS) estimation procedure of Im and Schmidt (2008) in order to improve the power

of the traditional Dickey-Fuller (1979, DF) unit root tests. We refer to this test as the RALS-DF

unit root test since it is an extension of the traditional DF test. The RALS procedure utilizes the

information that exists when the errors in the testing equation exhibit any departures from

normality, such as non-linearity, asymmetry, or fat-tailed distributions. The underlying idea of

the RALS procedure is appealing because it is intuitive and easy to implement. If the errors are

non-normal, the higher moments of the residuals contain the information on the nature of the

non-normality. The RALS procedure conveniently utilizes these moments in a linear testing

equation without the need for a priori information on the nature of the non-normality, such as the

density function or the precise functional form of any non-linearity. The power gain over the

usual DF tests is considerable when the error term is asymmetric or has a fat-tailed distribution.

This essay extends the work of Im, Lee and Tieslau (2012), and considers the Lagrange

Multiplier (LM) version of the RALS unit root tests. We refer to them as the RALS-LM tests.

We provide the relevant asymptotic distribution of these new tests and their corresponding

critical values. The LM unit toot tests were initially suggested by Schmidt and Phillips (1992,

SP). To begin with, consider an unobserved components model,

1

(1.0.1)

The unit root null hypothesis implies , against the alternative that . Here, the

parameters and will denote level and deterministic trend, respectively, regardless of whether

contains a unit root ( = 1) or not. The key difference between the LM and the DF procedures

is found in the detrending method. For the LM version tests, the coefficients of the deterministic

trend components are estimated from the regression in differences of on with

. Denoting the maximum likelihood estimates (MLE) from the LM procedure as and , SP

(1992) suggest using the detrended form of ,

. (1.0.2)

On the other hand, the DF test is based on the estimates of the coefficients from the

regression of in levels on .1 The LM tests show improved power over the DF tests. This

essay shows that the same feature will carry-over to the RALS version tests; the RALS-LM tests

show improved power over the RALS-DF tests.

The main advantage of the LM tests of SP (1992) is that they are less sensitive to the

parameters related to structural changes. In particular, they are free of nuisance parameters in

models with level shift, as we will explain in more detail in the next section. However, the DF

version tests do not have this property. As such, there are operating advantages of using the LM

version of the unit root test for models with structural changes, and the same feature can be

1 We note that the GLS tests of Hwang and Schmidt (1996), and the DF-GLS tests of Elliott, Rothenberg, and Stock (1996) adopt a detrending method similar to that of the LM test. For the GLS tests, the coefficients of the deterministic trend components are estimated from the regression in quasi-differences of (= ) on (= ), where is a nuisance parameter that takes on some small value. The GLS tests of Hwang and Schmidt (1996) use a fixed value which is given a priori as a small value, such that = 0.02 and

. The DF-GLS tests search for the optimal small value of that maximizes the power under the local alternative. When is zero, these GLS-based tests are identical to the LM tests of SP (1992). In reality, the difference in the power of the LM tests and the GLS tests is not significant. The main source of the power gain for the GLS tests is its use of the LM type detrending procedure, although searching for the optimal value of can lead to a marginal improvement in power.

6

utilized in the RALS-LM tests with level-shifts, although it would be difficult to consider the

RALS-DF tests with breaks.

The remainder of the essay is organized as follows. In Section 1.1, we discuss the LM

procedure and propose the RALS-LM tests. In Section 1.2, we examine the size and power

properties and compare them with those of the RALS-DF tests. Section 1.3 provides concluding

remarks.

1.1. LM and RALS-LM Tests

We are interested in testing the unit root null hypothesis against the stationary

alternative hypothesis . We let denote the deterministic terms, including structural

changes, and rewrite the data generating process (DGP) in (1.0.1) as:

. (1.1.1)

For example, if , we have the usual no-break LM test of SP (1992). To consider a

model with level shift where a break occurs at , we may add a dummy variable, , where

if and if . Then, we have

. (1.1.2)

Again, the parameter is estimated from the regression in differences of on where

. Here, the estimated value of will denote the magnitude of the level shift in a

consistent manner, regardless of whether contains a unit root or not. We do not need to

assume that under the null, and the critical values of the test will not change for different

values of . More importantly, the LM tests will not depend on the nuisance parameter,

, which denotes the location of the break, as shown in Amsler and Lee (1995). As

such, the same critical values of the usual LM test (without breaks) can be used even in the

7

presence of multiple level shifts. By contrast, Perron’s (1989) tests with level shifts depend on λ,

and different critical values need to be obtained for all different combinations of the break

locations in the case of multiple level shifts.

While the dependency on might be matter of minor inconvenience for the exogenous

tests of Perron (1989), the issue becomes complicated in the case of endogenous-break unit root

tests for which the location of the break is determined from the data where the t-statistic on the

unit root hypothesis is minimized, or the F-statistic on the dummy coefficients is maximized.

The popular endogenous-break unit root tests based on the DF version models can exhibit

spurious rejections unless the parameters in , which denote the magnitude of the structural

breaks (either in level-shifts or trend-shifts), take on zero values. Such an approach leads to a

conceptual difficulty of not allowing for breaks under the null of a unit root. Therefore,

rejections of the unit root null hypothesis will not necessarily imply trend-stationarity since the

possibility of a unit root with break(s) still remains; see Nunes et al. (1997), and Lee and

Strazicich (2001), among others for details. The LM-based tests, on the other hand, are free of

this problem in models with level-shift. In light of this, the LM tests with two endogenous

breaks are considered in Lee and Strazicich (2003) who allow for breaks both under the null and

alternative hypotheses in a consistent manner.

It also is possible to allow for multiple breaks by employing additional dummy variables

for multiple level shifts with where for ,

and zero otherwise; is the number of structural breaks. The invariance feature of the LM tests

with level-shifts is useful for extending the univariate LM tests to a panel setting. To that end,

Im, Lee and Tieslau (2005) suggest panel LM unit root tests with level breaks. Without the

8

invariance feature of the LM tests, the panel test statistic would not be feasible since the test

would depend on the nuisance parameters indicating the location of the breaks. This is

particularly true when each cross-section unit is likely to experience a different number and

location of breaks.

This essay shows that the same invariance feature in the LM tests also will hold in the

RALS-LM tests with level-shifts. In this case, the RALS-LM tests with multiple level-shifts will

have the same distribution as the RALS-LM tests without breaks.2 This is one main advantage

of the RALS-LM tests. Without the invariance feature of the LM tests, it would be difficult to

construct valid critical values for RALS-based tests, since the RALS procedure will induce an

additional nuisance parameter. Thus, it would be extremely difficult to construct valid RALS-

DF tests with breaks.

We now explain details of the RALS-LM procedure. In general, following the LM

(score) principle, the LM unit root test statistic can be obtained from the following regression:

, (1.1.3)

where , ; is the vector of coefficients in the regression of on

, and is the restricted MLE of given by ; and, and denote the first

observation of and , respectively. To control for autocorrelated errors, one can include the

terms , in (1.1.3) , and the testing regression is given as:

. (1.1.4)

2 The invariance property of the LM tests does not hold in models with trend-shifts where is used. However, the LM based tests are much less sensitive to the parameters of trend-breaks than the DF version tests. For example, Nunes (2004) found that the critical values do not change much in the models with trend-shifts

and considered a method using the same critical values regardless of different values of . However, the LM tests still depend on the nuisance parameter in these models, and using the same critical values can lead to mild size distortions.

9

Then, the LM test statistic is given by:

-statistic testing the null hypothesis .

Next, we explain how to utilize the information on non-normal errors in order to improve upon

the power of the unit root test, making use of the RALS estimation procedure as suggested in Im

and Schmidt (2008), and Im, Lee and Tieslau (2012). To begin with, we define

, and . Suppose we have the

following moment conditions:

, (1.1.5)

where is a function defined as with , and is a

nonlinear function of the error term . Then the moment condition becomes:

, (1.1.6)

. (1.1.7)

The first part is the usual moment condition of least squares estimation and the second part

involves an additional moment conditions based on nonlinear functions of . We let denote

the residuals from the usual LM regression (1.1.4). Following Im and Schmidt (2008), we define

the following term:

, (1.1.8)

where , , and . Using ,

we define the augmented terms:

. (1.1.9)

The RALS-LM procedure involves augmenting the testing regression (1.1.4) with . The first

term in is associated with the moment condition , which is the condition

10

of no heteroskedasticity. This condition improves the efficiency of the estimator of when the

error terms are not symmetric. The second term in improves efficiency unless . It

is possible to use higher moments using with , and the properly

defined in (1.1.9) that corresponds to the higher moments. The additional efficiency gain is

expected, unless which holds only for the normal distribution.3 Thus, when

the distribution of the error term is not normal, one may increase efficiency by augmenting the

testing regression with , as follows:

. (1.1.10)

The RALS-LM statistic is obtained through the usual least squares estimation procedure applied

to (1.1.10). We denote the corresponding -statistic for as . We adopt Assumption 1

and Assumption 2 of Im, Lee and Tieslau (2012) for the error term , and in (1.1.5),

respectively. Then it can be shown that the asymptotic distribution of is given as

follows.

Lemma 1. Suppose that we consider the usual -statistic on in equation (1.1.10). Then,

under the null, the limiting distribution of the RALS-LM t-statistic can be derived as

, (1.1.11)

where denotes the limiting distribution of the t-statistic for the usual LM estimator in

regression (1.1.4), and is the correlation between and

, (1.1.12)

3 However, we do not pursue this direction further and leave it as future research. This extension requires the assumption that the higher moments exist. In any case, the power gain is already significant enough when using the augmented terms in (1.1.9).

11

where

, , and .

PROOF:

Consider the regression (1.1.10). We let , where , and

. Following Theorem 2 in Hansen (1995), we have

From the moment condition , and , we have

Since , , , then we

have

Apply the lemma from Hansen (1995),

where . Additionally, we have

12

and . The test statistics under the null of can be obtained as

The null holds when . Then we obtain

□

These results are essentially similar to those given in Im, Lee and Tieslau (2012). Also,

the RALS-LM tests are asymptotically identical to the GMM estimators using the same moment

conditions in (1.1.6) and (1.1.7). It is interesting to see that the limiting distribution of is

similar to that of the unit root tests with stationary covariates, as advocated by Hansen (1995).4

The difference is how the parameter is estimated. We have a special case of Hansen’s models

and can be estimated by:

(1.1.13)

where is the usual estimate of the error variance in the LM regression (1.1.4), and is the

estimate of the error variance in the RALS-LM regression in (1.1.10).

Note that the asymptotic distribution of the RALS-LM test statistic does not

depend on the break location parameter in the model with level-shifts, following the results of

Amsler and Lee (1995). Thus, we do not need to simulate new critical values, regardless of the

number of level-shifts and all possible different combinations of break locations. From a

4 A similar asymptotic result also is advocated in Guo and Phillips (1998, 2001).

13

practical perspective, it likely would be infeasible to obtain all possible different critical values

corresponding to different break locations and values of . For a finite number of level-shifts,

we only need one set of critical values since they are asymptotically invariant to both the break

magnitude and location.

Note that when , we have , so that the critical value for the usual

LM test can be used. In Table 1.1, we report the asymptotic critical values of the RALS-LM

tests, for different values of and = 50, 100, 300 and 1000, respectively. All

of these critical values are obtained via Monte Carlo simulations using 100,000 replications.

These critical values can be used even when multiple level breaks occur in the data. To see this

we provide the empirical critical values of the RALS-LM tests when the number of level-shifts is

1 and 2. The results in Table 1.2 are virtually identical to the critical values of the RALS-LM

tests without breaks, as reported in Table 1.1.

1.2. Simulations

In this section, we investigate the finite small sample properties of the RALS-LM unit

root tests. Our goal is to verify the theoretical results presented above and examine the

performance of the tests. Pseudo-iid random numbers were generated using the RATS

procedure and all results were obtained via simulations in WinRATS 7.2. The DGP

was given in (1.1.2), and the initial values and are assumed to be random with mean zero

and variance 1. In order to examine the power when non-normal error exists, we consider seven

types of non-normal errors which include: (i) a chi-square distribution with , and

(ii) a t-distribution with . For purposes of comparison, we also examined the case

when the error term follows a standard normal distribution. The size and power property are

14

examined with two different DGPs; (a) no break with and , and (b) one

level shift with and . We also let denote the fraction of the

series before the break occurs at . For all of these cases, we have used the same

critical values in Table 1.1 of the usual RALS-LM tests without breaks. All simulation results

are calculated using 10,000 replications for the sample size, 100 by using the 5%

significance level.5

In Table 1.3, we report the size and power properties of the RALS-LM tests, and compare

them with the usual LM tests, the DF tests and the RALS-DF tests. We begin by examining the

model with no breaks. From Panel A in Table 1.3, we observe that, in all cases, none of the three

tests shows any serious size distortions. The RALS-LM tests show significantly improved power

over the usual LM tests when the errors are non-normal with either a chi-square or t-distribution.

The results in Panel C for the size-adjusted power are more relevant. The gain in power of the

RALS-LM tests is greater when the degrees of freedom of the chi-square distribution is smaller,

implying more asymmetric patterns of the error distribution. For example, when and the

error term follows a distribution, the size-adjusted power of the RALS-LM test is 0.976,

while the power of the usual LM test is 0.294 (and 0.187 for the DF test). Also, the gain is larger

when the degrees of freedom of the t-distribution is smaller, implying fatter-tails of the error

term.6 In Figure 1.1, we have provided a graph to show these results.

5 Results for the larger sample sizes with 300 and 1,000 are omitted. They show a similar pattern with greatly improved power properties. They are available with upon request. 6 The question of interest is the effect of using the estimated values of in (1.1.13) on the size and power property of the tests. The true value of is unknown, but it depends on the type and degree of non-normal errors. It seems clear that the size property is fair in all cases that we examined. The power gain would be larger when the value of is small. This occurs when the degrees of freedom of the chi-square distribution is smaller, implying more asymmetric patterns, and when the degrees of freedom of the t-distribution is smaller, implying fatter-tails. Our simulation results are consistent with our expectations.

15

When the error term follows a normal distribution, the LM tests are more powerful than

the RALS-LM tests. However, the difference in power is rather small. These results prove that

the RALS-LM tests have good size and power properties even when the sample size is relatively

small. In Panel D of Table 1.3, we report the size and size-adjusted power of the RALS-DF tests

of Im, Lee and Tieslau (2012). It seems clear that the RALS-LM tests are generally more

powerful than the RALS-DF tests. The difference is as expected since the LM tests are usually

more powerful than the DF tests.

Next, we examine the property of these tests when the DGP includes a structural break

where the size and power properties are examined for and . In this case, it is

not useful to consider the RALS-DF test since the distribution of the test depends on . The

results for the RALS-LM and traditional LM tests are presented in Table 1.4. Note that we use

the same critical values for the traditional LM tests without breaks and the RALS-LM tests

without breaks. Again, we do not observe any significant size distortions under the null, even

when the critical values of the tests without breaks are used for the models with breaks.

Regardless of the locations of breaks with either , or , the results on the size,

power and size-adjusted power do not change appreciably. This outcome clearly shows the

invariance results for both the LM and the RALS-LM tests. Also, we observe significant power

gains for the RALS-LM tests when the error term follows a non-normal distribution.

1.3. Concluding Remarks

This essay develops new RALS based LM unit root tests. These new RALS-LM tests

show improved power gains over the corresponding RALS-DF tests, and the power of both

RALS-DF and RALS-LM tests can increase drastically when the error term is highly asymmetric

16

or has fat-tails with unknown forms of non-normal distributions. Also, the RALS-LM tests have

the feature that they are invariant to the nuisance parameter in the models with multiple level

shifts, and it is expected that they can be more useful in extended models with other types of

structural changes.

Overall, we conclude that the RALS-LM tests show improved performance over the

corresponding RALS-DF tests. However, we should note that the power gain of the LM version

tests (and also the DF-GLS version tests) will disappear when the initial value is large. In such

cases, the RALS-DF version tests are more powerful than the RALS-LM version tests. As such,

one may consider a fair balance between the RALS-LM and the RALS-DF tests in the presence

of non-normal errors; it is incorrect to say that one version would dominate uniformly over the

other. Clearly, the main advantage of the RALS-LM tests lies in the invariance feature that the

distribution does not depend on the nuisance parameter in the presence of level-breaks, and that

they are less sensitive in other extended break models. On the other hand, the RALS-DF version

tests can be more useful in standard models without breaks, especially when the initial value is

large.

17

Table 1.1 Critical Values of RALS-LM Test with No Break

%

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

50

1 –2.333 –2.882 –3.089 –3.213 –3.314 –3.401 –3.477 –3.543 –3.599 –3.650 –3.707

5 –1.640 –2.216 –2.426 –2.572 –2.689 –2.783 –2.862 –2.928 –2.992 –3.045 –3.089

10 –1.286 –1.864 –2.081 –2.235 –2.358 –2.459 –2.545 –2.619 –2.684 –2.740 –2.793

100

1 –2.322 –2.875 –3.069 –3.200 –3.301 –3.378 –3.446 –3.499 –3.551 –3.595 –3.625

5 –1.646 –2.216 –2.422 –2.562 –2.672 –2.764 –2.840 –2.903 –2.958 –3.010 –3.062

10 –1.283 –1.861 –2.075 –2.230 –2.352 –2.450 –2.535 –2.606 –2.665 –2.719 –2.772

300

1 –2.332 –2.874 –3.057 –3.175 –3.265 –3.347 –3.403 –3.455 –3.509 –3.553 –3.596

5 –1.642 –2.211 –2.413 –2.554 –2.664 –2.755 –2.830 –2.893 –2.944 –2.992 –3.028

10 –1.276 –1.856 –2.071 –2.231 –2.351 –2.450 –2.532 –2.603 –2.664 –2.711 –2.758

1000

1 –2.345 –2.892 –3.080 –3.205 –3.299 –3.374 –3.428 –3.474 –3.510 –3.538 –3.570

5 –1.652 –2.223 –2.428 –2.568 –2.677 –2.761 –2.836 –2.897 –2.947 –2.990 –3.031

10 –1.291 –1.871 –2.083 –2.234 –2.352 –2.451 –2.535 –2.605 –2.667 –2.715 –2.755

18

Table 1.2 Critical Values of RALS-LM Test with Level Break

%

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

50

1

1 –2.333 –2.886 –3.077 –3.208 –3.309 –3.398 –3.478 –3.543 –3.605 –3.653 –3.711

5 –1.640 –2.216 –2.422 –2.570 –2.688 –2.784 –2.862 –2.932 –2.990 –3.044 –3.094

10 –1.286 –1.863 –2.081 –2.237 –2.358 –2.461 –2.546 –2.619 –2.683 –2.742 –2.794

2

1 –2.333 –2.891 –3.078 –3.209 –3.309 –3.398 –3.467 –3.532 –3.587 –3.646 –3.695

5 –1.640 –2.220 –2.427 –2.575 –2.696 –2.791 –2.867 –2.933 –2.991 –3.044 –3.100

10 –1.286 –1.866 –2.080 –2.235 –2.358 –2.460 –2.546 –2.622 –2.687 –2.743 –2.794

100

1

1 –2.322 –2.876 –3.066 –3.203 –3.307 –3.383 –3.456 –3.515 –3.560 –3.609 –3.637

5 –1.646 –2.218 –2.421 –2.563 –2.675 –2.761 –2.839 –2.903 –2.963 –3.011 –3.062

10 –1.283 –1.861 –2.076 –2.228 –2.351 –2.451 –2.535 –2.607 –2.667 –2.720 –2.771

2

1 –2.322 –2.877 –3.066 –3.204 –3.295 –3.372 –3.444 –3.497 –3.554 –3.583 –3.636

5 –1.646 –2.220 –2.422 –2.566 –2.677 –2.767 –2.842 –2.908 –2.962 –3.013 –3.056

10 –1.283 –1.862 –2.076 –2.231 –2.349 –2.450 –2.533 –2.604 –2.662 –2.717 –2.771

300

1

1 –2.332 –2.875 –3.055 –3.175 –3.270 –3.344 –3.402 –3.458 –3.505 –3.545 –3.588

5 –1.642 –2.213 –2.413 –2.557 –2.666 –2.757 –2.833 –2.892 –2.944 –2.991 –3.028

10 –1.276 –1.856 –2.070 –2.229 –2.349 –2.450 –2.531 –2.601 –2.660 –2.711 –2.757

2

1 –2.332 –2.875 –3.057 –3.175 –3.273 –3.353 –3.410 –3.457 –3.506 –3.554 –3.593

5 –1.642 –2.212 –2.414 –2.556 –2.664 –2.754 –2.831 –2.894 –2.945 –2.990 –3.023

10 –1.276 –1.856 –2.072 –2.229 –2.349 –2.449 –2.532 –2.604 –2.664 –2.709 –2.756

1000

1

1 –2.345 –2.895 –3.080 –3.204 –3.296 –3.376 –3.427 –3.470 –3.504 –3.536 –3.565

5 –1.652 –2.223 –2.428 –2.570 –2.677 –2.763 –2.836 –2.896 –2.948 –2.993 –3.029

10 –1.291 –1.870 –2.082 –2.234 –2.352 –2.452 –2.534 –2.607 –2.667 –2.715 –2.755

2

1 –2.345 –2.895 –3.077 –3.202 –3.300 –3.375 –3.427 –3.470 –3.510 –3.544 –3.571

5 –1.652 –2.221 –2.427 –2.567 –2.675 –2.762 –2.836 –2.897 –2.950 –2.990 –3.031

10 –1.291 –1.870 –2.082 –2.234 –2.353 –2.452 –2.536 –2.606 –2.665 –2.715 –2.756

Note: For = 1, the break is assumed at the middle. For = 2, the breaks are assumed at the one third and two thirds.

19

Table 1.3 Size, Power, and Size-Adjusted Power with No Break ( = 100)

Tests Distribution of the Error Term

N(0,1)

Panel A. Size

1

RALS-LM 0.057 0.059 0.060 0.064 0.045 0.042 0.052 0.086

LM 0.038 0.042 0.044 0.047 0.037 0.043 0.046 0.052

DF 0.046 0.051 0.046 0.049 0.053 0.049 0.052 0.051

Panel B. Power

0.8

RALS-LM 0.997 0.995 0.991 0.985 0.948 0.887 0.856 0.780

LM 0.770 0.757 0.763 0.765 0.789 0.755 0.762 0.754

DF 0.656 0.648 0.654 0.650 0.637 0.651 0.645 0.643

0.9

RALS-LM 0.977 0.929 0.870 0.809 0.664 0.474 0.393 0.341

LM 0.244 0.254 0.255 0.254 0.236 0.253 0.248 0.268

DF 0.170 0.181 0.180 0.182 0.161 0.180 0.183 0.193

Panel C. Size-Adjusted Power

0.8

RALS-LM 0.998 0.996 0.992 0.985 0.962 0.918 0.864 0.659

LM 0.817 0.789 0.787 0.776 0.848 0.787 0.779 0.745

DF 0.688 0.645 0.670 0.654 0.609 0.657 0.635 0.638

0.9

RALS-LM 0.976 0.925 0.858 0.784 0.705 0.515 0.389 0.227

LM 0.294 0.286 0.282 0.266 0.296 0.282 0.265 0.260

DF 0.187 0.178 0.190 0.185 0.150 0.183 0.177 0.190

Panel D. Size and Size-Adjusted Power of RALS-DF Tests

1.0 RALS-DF 0.049 0.059 0.062 0.068 0.047 0.053 0.056 0.054

0.9 RALS-DF 0.988 0.927 0.831 0.716 0.633 0.384 0.292 0.191

20

Table 1.4. Size, Power, and Size-Adjusted Power with Level Break ( = 100)

Tests Distribution of the Error Term

N(0,1)

Panel A. Size

1

0.25 RALS-LM 0.056 0.054 0.057 0.062 0.046 0.044 0.051 0.083

LM 0.040 0.046 0.045 0.046 0.037 0.043 0.047 0.053

0.50 RALS-LM 0.055 0.056 0.057 0.062 0.046 0.041 0.051 0.083

LM 0.040 0.045 0.043 0.047 0.035 0.041 0.049 0.052

Panel B. Power

0.8

0.25 RALS-LM 0.996 0.993 0.987 0.979 0.934 0.871 0.833 0.753

LM 0.743 0.730 0.735 0.733 0.764 0.727 0.732 0.724

0.50 RALS-LM 0.996 0.993 0.987 0.978 0.934 0.869 0.832 0.755

LM 0.739 0.727 0.738 0.739 0.761 0.731 0.732 0.726

0.9

0.25 RALS-LM 0.971 0.919 0.851 0.788 0.653 0.463 0.385 0.327

LM 0.235 0.248 0.251 0.250 0.227 0.247 0.246 0.260

0.50 RALS-LM 0.971 0.918 0.853 0.786 0.653 0.462 0.389 0.329

LM 0.235 0.249 0.251 0.249 0.231 0.243 0.246 0.256

Panel C. Size-Adjusted Power

0.8

0.25 RALS-LM 0.997 0.994 0.989 0.979 0.951 0.898 0.839 0.643

LM 0.785 0.748 0.758 0.752 0.822 0.760 0.744 0.715

0.50 RALS-LM 0.996 0.994 0.988 0.978 0.950 0.901 0.843 0.645

LM 0.788 0.758 0.768 0.758 0.829 0.770 0.737 0.719

0.9

0.25 RALS-LM 0.971 0.918 0.842 0.766 0.688 0.498 0.379 0.228

LM 0.281 0.266 0.272 0.266 0.285 0.275 0.257 0.252

0.50 RALS-LM 0.970 0.915 0.842 0.761 0.689 0.508 0.388 0.235

LM 0.275 0.273 0.277 0.262 0.293 0.283 0.251 0.250

21

Figure 1.1 Size-Adjusted Power, T = 100

Note: Size adjusted power of no break tests with = 0.9.

0

0.2

0.4

0.6

0.8

1

1.2

chi-1 chi-2 chi-3 chi-4 t-2 t-3 t-4 Normal

Power

RALS-LM

RALS-DF

LM

DF

22

CHAPTER 2

THE PREBISCH-SINGER HYPOTHESIS AND RELATIVE COMMODITY PRICES: FURTUER EVIDENCE ON BREAKS AND TREND SHIFTS

The Prebisch-Singer hypothesis (PSH) postulates a secular decline in commodity prices

relative to manufactured goods in the long-run (Prebisch, 1950; Singer, 1950). The basis for the

declining trend includes a low income elasticity of primary commodities, productivity

differentials between industrial and commodity producing countries, and asymmetric market

structures (oligopolistic rents in manufacturing versus zero profit competition in primary goods).

The policy implication associated with the PSH for developing countries, to the extent they

export primary commodities and import manufactured goods, is deterioration in their net barter

terms of trade. To counter this economic outcome, developing countries may pursue policies of

export diversification away from primary commodities or import substitution to increase the

domestic production of manufactured goods. Indeed, in the context of development efforts in

Africa, Deaton (1999) notes that countries that rely on primary commodity exports need to

understand the nature of commodity prices in order to design effective development and

macroeconomic policy.

Early studies by Spraos (1980), Sapsford (1985), Thirwall and Bergevin (1985), and

Grilli and Yang (1988) of the PSH focused on examining the sign and significance of the

estimated trend term based on the assumption that the logarithm of relative commodity prices is

stationary, typically around a linear trend, with the results supportive of the PSH. However, this

seemingly simple issue is more complicated than it appears due to the possibility of non-

1

stationarity of the data. As such, more recently, the research focus has been on determining

whether relative commodity prices are trend stationary (i.e. deterministic trend) or difference

stationary (i.e. stochastic trend). The distinction between trend stationary and difference

stationary processes is important. If relative commodity prices contain a unit root, the standard

method of least squares to test for the presence of a trend will suffer from severe size distortion.

On the other hand, if relative commodity prices are generated by a trend stationary process, yet

modeled as a difference stationary, the test will be inefficient and will lack power relative to the

trend stationary process (Perron and Yabu, 2009). Studies by Bleaney and Greenaway (1993),

Newbold and Vougas (1996), and Kim et al. (2003) have noticed relative commodity prices may

be characterized as difference stationary and the relative commodity prices evolve according to a

stochastic trend instead of a deterministic trend, rendering less support for the PSH.

One additional complication is how to deal with structural breaks in examining relative

commodity prices. In light of the study by Perron (1989), which demonstrated that if a structural

break is ignored the power of unit root tests is lowered, more recent studies by Cuddington and

Urzúa (1989), Powell (1991), and Cuddington (1992) introduce structural breaks in testing for

the PSH. However, given that break locations are not known, a number of recent studies by

Leon and Soto (1997), Zanias (2005), and Kellard and Wohar (2006), utilize unit root tests with

allowance for endogenously determined structural breaks.1 However, Ghoshray (2011) notes the

spurious rejection problem of the endogenous unit root tests that most of these studies employed.

The spurious rejection problem occurs when endogenous break unit root tests attempt to

1 Most popular endogenous break unit root tests include Zivot and Andrews (1992), Lumsdaine and Papell (1997), Perron (1997), Vogelsang and Perron (1998), and Lee and Strazicich (2003). Those tests feature searching for the breaks using a grid search. The location of a break is determined at the place where the t-statistic on the unit root hypothesis is minimized or the t-statistic for the break coefficient is maximized.

24

eliminate the dependency on the nuisance parameter by assuming that breaks are absent under

the null. Then, as pointed out by Nunes et al. (1997) and Lee and Strazicich (2001), assuming

away the nuisance parameter leads to a serious size distortion resulting in spurious rejections

under the null hypothesis. The LM test suggested by Lee and Strazicich (2003) allows for

structural breaks under the null hypothesis and do not suffer from the spurious rejection of the

null hypothesis. As such, Ghoshray (2011) adopts the LM endogenous tests of Lee and

Strazicich (2003) to provide different test results; see also Harvey et al. (2010), and Kejriwal et

al. (2012). These studies provide mixed evidence regarding the PSH; however, the studies

generally raise doubts as to whether relative commodity prices are best represented by a single

downward trend or a shifting trend changing over time.

In this essay, we re-examine the path of relative primary commodity prices. For this, we

suggest a new Residual Augmented Least Squares–Lagrange Multiplier (RALS-LM) unit root

tests with trend breaks and non-normal errors. The motivation of the new approach is to utilize

all possible information to maximize the power of the tests but to make the test free of nuisance

parameters. In particular, we try to utilize the information on non-normal errors. Non-normal

errors might have been ignored in the literature but they can provide valuable information for

improving the power. Specifically, we adopt the LM unit root tests with trend breaks combined

with the RALS method to utilize the information contained in non-normal errors. The RALS

methodology was initially suggested by Im and Schmidt (2008). The Essay 1 develops the

RALS procedure for the LM tests, but those tests do not allow for trend breaks. We showed that

the RALS-LM tests gain improved power with non-normal errors and are fairly robust to some

forms of non-linearity. However, these tests also lose power when existing trend breaks are not

25

taken into account. In our analysis of the path of relative primary commodity prices, it seems

very likely that trend-shifts might have occurred in the data; see Figure 2.1, for example. Thus,

it is necessary to develop new tests that allow for possible trend-shifts when the RALS method is

adopted. In doing so, we adopt a procedure to make the resulting test free of the nuisance

parameter problem which trend-shifts induce otherwise. We will show that the resulting RALS-

LM tests with breaks become much more powerful in the presence of non-normal errors.

Based on the discussion mentioned above, it is reasonable to believe the results given by

the new RALS-LM tests with trend breaks will provide more accurate information on the path of

relative commodity prices. We apply the new tests to the recently extended Grilli and Yang

index which contains 24 time series of relative primary commodities prices over the period 1900-

2007. We find more commodity price time series are stationary by utilizing possible non-normal

errors which have been ignored in previous studies.

The remainder of the study is organized as follows. Section 2.1 discusses the RALS-LM

tests with level and trend breaks. In section 2.2, we provide simulation results about the small

sample performance of the new test. Section 2.3 presents the data and the empirical results.

Concluding remarks are given in Section 2.4.

2.1. Econometric Methodology

In this study, to examine relative commodity prices, we wish to employ the most reliable

and powerful tests that utilize all major factors. To begin with, we consider the following data

generating process (DGP) based on the unobserved component representation:

, (2.1.1)

26

where contains exogenous variables. The unit root null hypothesis is . The level-shift

only, or “crash,” model can be described by , where for and

zero otherwise, and stands for the time period of the break. The trend-break model can be

described by , where for , and zero otherwise. A

more general model which allows for both level-shift and trend-shift can be described as

. This general model will be the focus of our study. To consider multiple

breaks, we can include additional dummy variables such that:

, (2.1.2)

where for , , and zero otherwise, and for

and zero otherwise. Following the LM (score) principle, we impose the null

restriction and consider in the first step the following regression in differences:

, (2.1.3)

where , . The unit root test statistics are then obtained from the

following regression:

, (2.1.4)

where denotes the de-trended series

. (2.1.5)

We let be the -statistic for from (2.1.4). Here, is the coefficient in the

regression of on in (2.1.3), and is the restricted MLE of : that is, .

Subtracting in (2.1.5) makes the initial value of the de-trended series to begin at zero with

, but letting leads to the same result. It is important to note that in the de-trending

procedure (2.1.5), the coefficient was obtained in regression (2.1.3) using first differenced

27

data. Thus, the de-trending parameters are estimated in the first step regression in differences.

Through this channel the dependency on nuisance parameters is removed in the crash model.

However, the dependency on nuisance parameters is not removed with this de-trending

procedure in the model with trend breaks. In such cases, it will be difficult to combine these LM

tests with the RALS procedure which will induce a new parameter. Indeed, the dependency of

the tests on the nuisance parameter in the trend-shift models has been an issue in the literature,

since it poses a difficulty in developing more extended tests. The same problem occurs in our

case when we wish to utilize information on non-normal errors. As such, we consider a simple

transformation which can make the LM unit root test statistic free of the dependency on the

break location as in Lee et al. (2012). Specifically, we can see that usual LM tests for the model

with trend-shifts will depend on , which denotes the fraction of sub-samples in each regime

such that , , , and .

However, the following transformation can remove the dependency on the nuisance

parameter:

(2.1.6)

We then replace in the testing regression (2.1.4) with as follows:

. (2.1.7)

We denote as the -statistic for from (2.1.7). Then, the asymptotic distributions of

the test statistic will be invariant to the nuisance parameter .

28

, (2.1.8)

where is the projection of the process on the orthogonal complement of the space

spanned by the trend break function , as defined over the interval , where

, and is a Wiener process for .

Following the transformation, the asymptotic distribution of depends only on the

number of trend breaks, since the distribution is given as the sum of independent

stochastic terms. With one trend-break ( ), the distribution of is the same as that of the

untransformed test using , regardless of the initial location of the break(s).

Similarly, with two trend-breaks ( ), the distribution of is the same as that of the

untransformed test using and . In general, for the case of multiple

breaks, the same analogy holds: the distribution of is the same as that of the untransformed

test using , . Therefore, we do not need to simulate numerous

critical values at all possible break point combinations. The critical values of are reported

in Lee et al. (2012), but instead of using these tests, we move on to the next step.

To improve the power of the LM statistic , we adopt the procedure to utilize the

information on non-normal errors. We adopt the RALS method as in Im et al. (2012). The

RALS procedure is to augment the following term to the testing regression (2.1.7),

, (2.1.9)

29

where , , and . To capture the information

of non-normal errors, we let , which involves the second and third moments of .

Then, letting , the augmented term can be given as

. (2.1.10)

The first term in is associated with the moment condition ,

which is the condition of no heteroskedasticity. The second term in improves efficiency

unless . This condition improves the efficiency of the estimator of when the error

terms are not symmetric. In general, knowledge of higher moments are uninformative if

. This is the redundancy condition. The normal distribution is the only

distribution that satisfies the redundancy condition. However, if the distribution of the error term

is not normal, the condition is not satisfied. In such cases, one may increase efficiency by

augmenting the testing regression (2.1.7) with . Then the transformed RALS-LM test statistic

with trend-breaks is obtained from the regression

.

(2.1.11)

We denote the corresponding -statistic for as . Then it can be shown that the

asymptotic distribution of is given as follows:

, (2.1.12)

where reflects the relative ratio of the variances of two error terms.

It is interesting to see that the limiting distribution is similar to the distribution of the unit

root tests with the stationary covariates of Hansen (1995). If the usual Dickey-Fuller (DF) tests

without breaks were used, the limiting distribution would be identical. However, one can hardly

30

combine the DF tests with breaks with the RALS procedure due to the dependency on the

parameters indicating the locations of break(s), . But, note that the asymptotic distribution of

the transformed RALS-LM test statistic no longer depends on the break location

parameter. Thus, we do not need to simulate new critical values for all the possible break

location combinations, and this paves the way for us to employ the RALS procedure in the

presence of trend-shifts. For a specific number of breaks, we just need one set of critical values

since the critical values are asymptotically invariant to the break location. We note in passing

that they are also invariant to the magnitude of breaks, contrary to some of popular endogenous

unit root tests; see Lee and Strazicich (2001, 2003) and Perron (2006). The critical values of the

transformed break RALS-LM unit root test with trend breaks are newly provided in Appendix

Table 1, for 1 and 2, 50, 100, 300 and 1000, and 0 to 1. All these critical values

are obtained via Monte Carlo simulations using 100,000 replications. These critical values can be

used when the break locations are known, or they are estimated consistently in a two-step

procedure.

The new unit root tests used in this study differs from those used in the previous studies

in several important aspects. First, we use linear unit root tests with structural breaks instead of

non-linear unit root tests. It is true that conventional linear unit root tests will lose power in the

presence of non-linearity; however, when dealing with real world data we rarely know the

specific form of the non-linearity. If the forms of non-linearity are unknown and mis-specified,

non-linearity tests are often less powerful than the DF type linear tests (see Choi and Moh,

2007). As our RALS-LM unit root tests are based on the linearized model specifications, the

31

tests will not be subject to this difficulty. Moreover, non-normality can possibly mimic some

unknown forms of non-linearity indirectly in a linear model framework.

Second, the new tests are free of the nuisance parameter and spurious rejection problems

although the distribution of unit root tests with structural breaks depends on the parameter(s)

indicating the location of break(s). Unlike the DF based tests, the usual LM tests do not suffer

from spurious rejections, but they still depend on the location parameters in the model with

trend-shifts. In such cases, they cannot be combined easily with the RALS procedure. As such,

we adopt a simple transformation of the data which eliminates dependency on the trend breaks.

As a result, the same critical values can be used at different break locations, even when the LM

tests are combined with the RALS procedure to capture non-normal errors at the same time.

Third, the new LM tests select the proper number of breaks determined from the data.

The endogenous break unit root tests used in several previous studies usually find and include

the number of breaks as pre-specified in the model. However, as discussed in Lee et al. (2012),

one or more unnecessary breaks will be included in the unit root test and as a result exhibits

lower power. Whether or not a structural break exists is an empirical issue that must be

determined from the data. As such, a two-step procedure can be more meaningful and the new

LM tests are more powerful compared with the tests used in previous studies. This approach is

favored in recent studies, given that the so-called endogenous unit root tests tend to exhibit

spurious rejection and/or nuisance parameter problems; see Perron (2006) and Lee et al. (2012).

The two-step procedure shows much improved performance in terms of both size and power

properties. In Appendix Table 2, we have examined the properties of the new RALS-LM tests.

We could observe that the performance of the tests under the null is fairly good in the presence

32

of various trend-shifts. Also, we observe a significant increase in power when the errors are non-

normal. Thus, we can achieve improved power by using the information on non-normal errors,

but without using any nonlinear functional form. The transformation procedure works

reasonably well.

2.2. Monte Carlo Experiment

We want to examine briefly the property of the transformed RALS-LM test using

different combinations of non-normal errors and break locations. Pseudo-iid random

numbers were generated using the RATS procedure and all results were obtained via

simulations using WinRATS (v. 8.1). The DGP was given as the form of (1). The initial values

and are assumed to be random. We consider the following non-normal errors: (i - iv)

distribution with 1 to 4, and (v - vii) -distribution with 2 to 4. We also include the

size and power for (viii) standard normal, transformed LM test, untransformed RALS-LM and

LM test as well as DF test for comparison. All simulation results are calculated using 10,000

simulations for sample size 100. 2 The size (frequency of rejections under the null when )

and power (frequency of rejections under the alternative when and ) of the tests

are examined using 5% critical values. The finite sample size and power property for

transformed RALS-LM test and comparing tests for 100 and 1 are presented in

Appendix Table 2.

2 Results for T = 300 and 1,000 are similar and available upon request from author.

33

We consider DGP with two different s: 0.25 and 0.5, but we use only one

set of critical values calculated based on 0.5. For all the five tests examined, we see little

size distortion whether 0.25 or 0.5. For the power property, we see large power gain when

RALS procedure is used than the corresponding non-RALS type test. When the non-normal error

becomes normal (degree of freedom becomes larger for the distribution and -distribution),

the power gain becomes less. One surprising result is the power for the RALS based tests are

larger than the corresponding non-RALS based test even when the error term follows a normal

distribution. To the contrary, the transformation procedure seems to make the tests less powerful

than the un-transformed tests. However, since the LM type of unit root test statistics are

dependent on the trend breaks, we need the transformation procedure to make the test valid.

Fortunately, for all the non-normal distributions of error term, the power gain from the RALS

procedure is much higher than the power loss from the transformation procedure.

2.3. Data and Empirical Results

In this study, we use recently extended Grilli and Yang annual data on prices for twenty-

four primary commodity prices spanning 1900-2007 provided by Pfaffenzeller et al. (2007). The

commodities examined are aluminum, banana, beef, cocoa, coffee, copper, cotton, hides, jute,

lamb, lead, maize, palm oil, rice, rubber, silver, sugar, tea, timber, tin, tobacco, wheat, wool and

zinc.

For each primary commodity , we deflate the nominal prices with the United Nations

Manufactures Unit Value Index (MUV) as follows:

34

. (2.3.1)

To examine the relative primary commodity prices in (2.3.1), we utilize the two-step LM

unit root test suggested by Lee et al. (2012)3 and three-step RALS-LM unit root test with trend

shifts. Throughout, we consider a model with at most two level and trend breaks.

The two-step LM test can be summarized as follows: In the first step, we set a maximum

structural break number (in this essay = 2) and apply the test to identify the break

locations, test the significance of each break, and simultaneously determine the optimal lags with

the corresponding number and location of breaks. If the null of no trend break is not rejected or

if the null of no trend break is rejected but one of the break dummy variables is not significant

based on the standard -test, we move to the beginning of the first step with the structural break

number equal to . This procedure continues until the break number becomes zero or all the

identified break dummy variables are significant. If the first step yields a break number equal to

zero, we use the usual no-break LM unit root test of Schmidt and Phillips (1992); if one or more

breaks are found, we use one break (or breaks) LM unit root test of Amsler and Lee (1995) and

Lee and Strazicich (2003) with the break number and location as well as the corresponding

optimal lags determined in the first step. Then, we obtain the LM test statistic, denoted as .

The first two steps of the RALS-LM test are the same as the two-step LM test. We use the

higher moment information obtained from the second step and augment it to the regression of the

two-step LM test as the third step of the RALS-LM test denoted as . While searching

for the optimal number of breaks, we use the grid search within 0.10-0.90 intervals of the whole

3 See also Yabu and Perron (2009), and Kejriwal and Perron (2010). These papers show consistently that the two-step based unit root tests have better size and power properties than endogenous unit root tests.

35

sample period so that each subsample before and after the breaks will have enough observations

to perform a valid test.4 The corresponding number of lags is chosen using a general to specific

approach with maximum lags equal to eight.5

As a prelude to the test of a unit root allowing for structural breaks, we examine the 24

primary commodity prices using Augmented Dickey-Fuller (ADF) test, LM test of Schmidt and

Phillips (1992) and RALS-LM test without breaks. We report the results in Table 2.1, labeled

, and , respectively. The preliminary results show that 12 relative

commodity prices reject a unit root using the ADF test at the 10% significance level; the

numbers for LM test and RALS-LM test are 9 and 14, respectively. The relative commodity

price series (cocoa, rubber, silver, wheat and zinc) reject the unit root using the RALS-LM test

but cannot reject a unit root using the LM test with all having low values,6 which indicate the

extra rejections come from using the non-normal information embedded in the error term. Out of

the 12 rejections using the ADF test, 11 of them are also rejected using the RALS-LM test. The

non-rejection of the maize price series may be a result of not accounting for breaks in the

intercept or trend.7

4 When using more than one break, we set the minimum length between two breaks to be at least 1/10 of observations for the same reason. 5 The general to specific method can be explained as follows: For each combination of breaks, we start with first eight lags, and examine the significance of the eighth lag (or ). If it is significant at 10% level (the absolute value of the t-statistics is greater than 1.645), we select eight lags as the optimal lags; if not, we try the first seven lags, and repeat the previous check. The searching procedure ends when the coefficient of the last lag , , is significant in which case we select the optimal lags to be ; otherwise, we select the optimal lags to be zero. 6 The of these five commodity prices are 0.619 (cocoa), 0.512 (rubber), 0.438 (silver), 0.630 (wheat), and 0.429 (zinc), respectively. The range for is [0, 1]; lower means the RALS procedure uses more information in the non-normal errors. 7 As we can see, after including one or two breaks in the transformed LM and RALS-LM tests, the unit root in the maize price series was rejected at the 1% significance level.

36

Tables 2.2 and 2.3 report unit root tests allowing for one and two trend breaks,

respectively with the test statistic using the transformed LM test and the test

statistic using transformed RALS-LM test. Using the one-break transformed LM and RALS-LM

tests, the number of rejections is 21 and 19, respectively, while the rejections in the

corresponding two-break tests in Table 2.3 are both 21. While it appears we obtain at least as

many trend stationary series when we add more trend breaks to the model, it is not necessarily

so. For example, we reject the null of a unit root in the lamb price series at the 5% significance

level in the no-break LM test, however, we cannot reject the null hypothesis of a unit root with

the one/two- break LM test; the same situation occurs with the cocoa, lamb, palm oil and silver

price series with the no-break LM test. Moreover, we can reject the null hypothesis of a unit root

for the palm oil price series at the 5% significance level using the ADF test, no-break LM test,

no-break RALS-LM test, and the one-break LM test; however, we cannot reject the null

hypothesis of a unit root in the one-break RALS-LM test, two-break LM test, and two-break

RALS-LM test.

Next, we apply our new two-step LM unit root test and the three-step RALS-LM unit root

test, the results of which are shown in Table 2.4. As discussed above, when using the two-step

and three-step tests, we select the optimal number of breaks using a procedure suggested

in Lee et al. (2012). Here, the optimal break number is determined using a general-to-specific

approach. The optimal break number is set to two if both trend break coefficients are significant

at the 10% level; if one of them is insignificant, we examine the trend break coefficient from the

one break test; if it is insignificant, the no break test is selected. For example, the second trend

break in the banana price series and the first trend break in the lead price series reveal that the

37

pre-specified two trend breaks are not significant at the 10% level, but the trend breaks for these

two series using the one-break test are significant. Thus, we select the optimal number of breaks

for the banana and lead price series to be one. The three relative commodity price series found

to be difference stationary are copper, lamb and palm oil.

Following Ghoshray (2011) and Kellard and Wohar (2006), we examine the prevalence

of trends in the primary commodity prices using the optimal breaks identified in Table 2.4. Panel

A of Table 2.5 displays the results of the estimated trend stationary models with one and two

structural breaks. A trend stationary model is estimated with the error term following an

ARMA(p, q) process. Out of the 21 trend stationary price series, 12 relative commodity prices

are found to exhibit a significant negative trend, though not necessarily for the entire sample.

The remaining 9 relative commodity price series have no evidence of a significant positive or

negative trend or show a mixture of positive trend and no trend. Panel B of Table 2.5 shows the

results of the estimated difference stationary models for those relative commodity price series

found to exhibit unit root behavior. A difference stationary model is estimated with the error

term following an ARMA (p, q) process. All three relative commodity price series considered

(copper, lamb, and palm oil) display a significant negative trend or a mixture of a significant

negative trend and positive trend. Then, we synthesize the results in Table 2.5 by constructing

(-), (+), and (.) that measure the prevalence of a negative trend, positive trend, and trendless

behavior, respectively. We define , where equals the number of years that

a statistically significant negative trend exists. Similarly, we define for a

significant positive trend, and for no trend. The results of the trend

prevalence among the relative commodity prices are shown in Table 2.6.

38

The prevalence of a negative trend is found in 15 commodities. That is, each of these

commodities displays at least one significant negative trend segment. Out of the 15 commodities,

7 commodities display a negative trend for more than 50% of the sample period. The prevalence

of a positive trend is found in 10 commodities, of which 4 commodities display a positive trend

for more than 50% of the sample period. However, the prevalence of trendless behavior is found

in 19 commodities, of which 8 commodities (aluminum, banana, copper, cotton, lead, palm oil,

silver and tin) display trendless behavior for more than 90% of the sample period.

Overall, the results suggest that the trend is variable, and there is no evidence of a single

negative trend. To visualize our empirical findings, we superimpose the level and trend breaks

identified by the two-break test in Table 2.5 and plot the relative commodity prices. Linear

trends are then estimated using OLS to connect the break points as shown in Figure 2.1. The

optimal break points identified in Table 2.4 are concentrated in two groups. Among the 46

breaks, 20 breaks show up between 1910-1930 and 11 breaks show up between 1970-1990,

which account for more than 67% of the all the identified breaks. Compared with past studies,

our findings provide even weaker evidence to support the PSH. 8

2.4. Concluding Remarks

This study employs newly developed LM unit root tests to determine whether relative

primary commodity prices contain stochastic trends. Unlike the endogenous break unit root test,

our two-step LM and three-step RALS-LM tests always include the appropriate number of

8 Persson and Teräsvirta (2003) and Balagtas and Holt (2009) have adopted a flexible model and estimated a number of smooth transition autoregressions (STARs). They also find limited support for the PSH.

39

breaks in the model. Simulation results show a power gain from the RALS procedure when the

error term follows a non-normal distribution. Since the unit root tests are free of spurious

rejections, as well as more powerful than those employed in previous studies, the rejection of the

null can be considered as genuine evidence of stationarity. The main findings of this study reveal

that 21 out of the 24 commodity prices are found to be stationary around a broken trend,

implying that shocks to these commodities tend to be transitory. Only three relative commodity

price series are found to be difference stationary. There are only 7 series in which the relative

commodity prices display negative trend more than 50% of the time period examined; however,

8 relative commodity prices display no significant positive or negative trend in more than 90% of

the time period examined. Compared with past studies, our findings provide even weaker

evidence to support the PSH. Also, we believe that in light of the significantly improved power,

the newly suggested RALS-LM tests with trend-shifts can be useful in other time series

applications in agricultural economics and related areas.

40

Table 2.1 Results using ADF Test and No-Break LM Unit Root Tests

ADF LM RALS-LM

Aluminum -3.180* 7 -3.520** -3.593*** 0.554 7

Banana -2.390 2 -1.547 -1.568 1.015 2

Beef -2.049 5 -1.909 -1.503 0.537 5

Cocoa -2.557 2 -2.250 -5.317*** 0.619 2

Coffee -3.315* 0 -3.353** -4.806*** 0.648 0

Copper -1.598 8 -1.983 -1.805 0.799 8

Cotton -2.975 2 -2.057 -2.422 0.893 2

Hides -3.925** 3 -3.322** -3.527** 0.830 3

Jute -3.285* 3 -3.044* -3.125** 0.949 3

Lamb -3.411* 4 -3.442** -3.398** 0.821 4

Lead -2.376 1 -1.858 -1.674 0.777 0

Maize -5.667*** 0 -1.817 -1.589 0.695 4

Palm oil -4.829*** 0 -4.082*** -3.035** 0.577 0

Rice -4.024** 7 -3.546** -4.249*** 0.840 7

Rubber -2.114 0 -2.212 -2.883** 0.521 0

Silver -2.242 2 -2.336 -4.434*** 0.438 2

Sugar -3.923** 2 -3.920*** -6.759*** 0.449 2

Tea -2.091 7 -2.207 -2.338 0.817 7

Timber -4.199*** 3 -3.532** -3.470** 0.938 3

Tin -2.355 0 -2.321 -2.023 0.759 0

Tobacco -1.414 4 -1.580 -1.721 0.911 4

Wheat -4.042*** 4 -1.503 -2.912** 0.630 6

Wool -3.093 4 -1.554 -1.763 0.819 4

Zinc -4.875*** 1 -2.353 -4.073*** 0.429 2

Notes: Since our LM test and RALS-LM test share the same procedure when searching for the optimal lags, we only report one time to save the space. is the optimal number of lagged first-differenced terms. , and denote the test statistics for the ADF test, LM test and RALS-LM test respectively. *, ** and *** denote the test statistic is significant at 10%, 5% and 1% levels, respectively.

41

Table 2.2 Results using Transformed One-Break LM and RALS-LM Tests

LM RALS-LM

Aluminum -4.847*** -6.105*** 0.404 1916 7

Banana -3.680* -4.016** 0.915 1966 6

Beef -4.511*** -3.110* 0.619 1972 1

Cocoa -1.755 -1.750 0.875 1976 2

Coffee -4.487*** -6.899*** 0.680 1975 0

Copper -3.203 -3.156 0.780 1969 6

Cotton -4.266** -4.691*** 0.929 1927 2

Hides -4.115** -3.925** 0.850 1932 3

Jute -4.715*** -4.584*** 1.012 1970 3

Lamb -2.940 -3.196 0.820 1974 4

Lead -3.450* -4.525*** 0.719 1978 1

Maize -6.009*** -5.829*** 0.591 1948 1

Palm oil -4.123** -2.871 0.540 1983 0

Rice -4.494*** -4.039** 0.899 1972 1

Rubber -5.512*** -8.208*** 0.553 1912 1

Silver -3.533* -1.010 0.508 1979 6

Sugar -4.135** -4.800*** 0.501 1920 6

Tea -3.905** -3.702** 0.871 1966 7

Timber -4.307*** -4.339*** 0.869 1979 3

Tin -4.088** -5.605*** 0.798 1984 0

Tobacco -3.596* -3.808** 0.904 1988 4

Wheat -4.867*** -5.649*** 0.666 1919 6

Wool -5.257*** -4.616*** 0.793 1956 3

Zinc -5.587*** -7.502*** 0.561 1914 1

Notes: Since our LM test and RALS-LM test share the same procedure when searching for the break points and the corresponding optimal lags, we only report one time to save the space. is the optimal number of lagged first-differenced terms. denotes the estimated break point. and

denote the test statistics for the transformed LM test and RALS-LM test respectively. *, ** and *** denote the test statistic is significant at 10%, 5% and 1% levels, respectively.

42

Table 2.3 Results using Transformed Two-Break LM and RALS-LM Tests

LM RALS-LM

Aluminum -6.995*** -6.732*** 0.527 1915 1927 7

Banana -5.274*** -5.789*** 0.903 1949 1984n 8

Beef -6.604*** -5.109*** 0.888 1956 1972 8

Cocoa -6.793*** -10.475*** 0.514 1945 1955 1

Coffee -6.056*** -8.018*** 0.651 1974 1987 3

Copper -3.449 -3.109 0.843 1915 1925 1

Cotton -8.361*** -8.641*** 0.886 1929 1951 1

Hides -7.001*** -7.496*** 0.842 1919 1930 0

Jute -5.494*** -5.487*** 0.991 1930 1965 3

Lamb -3.740 -3.269 0.899 1936 1956 4

Lead -4.186* -5.539*** 0.688 1920n 1978 1

Maize -7.776*** -6.226*** 0.820 1918 1947 3

Palm oil -3.427 -1.108 0.779 1917 1927 8

Rice -6.500*** -6.294*** 0.773 1915 1980 1

Rubber -8.242*** -9.396*** 0.738 1911 1924 2

Silver -10.254*** -13.900*** 0.475 1972 1983 1

Sugar -7.083*** -14.038*** 0.313 1923 1936 1

Tea -5.105*** -4.414** 0.904 1919 1966 7

Timber -7.518*** -7.799*** 0.930 1966 1979 1

Tin -5.750*** -6.162*** 0.866 1972 1984 1

Tobacco -4.766** -4.268** 0.955 1915 1988 4

Wheat -7.989*** -8.748*** 0.750 1919 1945 3

Wool -5.640*** -4.334** 0.792 1913 1956 3

Zinc -5.904*** -6.332*** 0.449 1917 1988 1

Notes: is the optimal number of lagged first-differenced terms. denotes the estimated break point. denotes that the identified break point was not significant at the 10% level. and

denote the test statistics for the transformed LM test and RALS-LM test respectively. *, ** and *** denote the test statistic is significant at 10%, 5% and 1% levels, respectively.

43

Table 2.4 Results using Two-Step LM and Three-Step RALS-LM Tests

LM RALS-LM

Aluminum -6.995*** -6.732*** 0.527 1915 1927 7

Banana -3.680* -4.016** 0.915 1966 6

Beef -6.604*** -5.109*** 0.888 1956 1972 8

Cocoa -6.793*** -10.475*** 0.514 1945 1955 1

Coffee -6.056*** -8.018*** 0.651 1974 1987 3

Copper -3.449 -3.109 0.843 1915 1925 1

Cotton -8.361*** -8.641*** 0.886 1929 1951 1

Hides -7.001*** -7.496*** 0.842 1919 1930 0

Jute -5.494*** -5.487*** 0.991 1930 1965 3

Lamb -3.740 -3.269 0.899 1936 1956 4

Lead -3.450* -4.525*** 0.719 1978 1

Maize -7.776*** -6.226*** 0.820 1918 1947 3

Palm oil -3.427 -1.108 0.779 1917 1927 8

Rice -6.500*** -6.294*** 0.773 1915 1980 1

Rubber -8.242*** -9.396*** 0.738 1911 1924 2

Silver -10.254*** -13.900*** 0.475 1972 1983 1

Sugar -7.083*** -14.038*** 0.313 1923 1936 1

Tea -5.105*** -4.414** 0.904 1919 1966 7

Timber -7.518*** -7.799*** 0.930 1966 1979 1

Tin -5.750*** -6.162*** 0.866 1972 1984 1

Tobacco -4.766** -4.268** 0.955 1915 1988 4

Wheat -7.989*** -8.748*** 0.750 1919 1945 3

Wool -5.640*** -4.334** 0.792 1913 1956 3

Zinc -5.904*** -6.332*** 0.449 1917 1988 1

Notes: is the optimal number of lagged first-differenced terms. denotes the estimated break point. and denote the test statistics for the two-step LM test and three-step RALS-LM test respectively. *, ** and *** denote the test statistic is significant at 10%, 5% and 1% levels, respectively.

44

Table 2.5 Estimated Trend Stationary and Difference Stationary Models

Regime 1 Regime 2 Regime 3 ARMA

Panel A. Estimated trend stationary models with breaks

Aluminum 1915 1927 0.041(0.102) -0.164(-0.874) -0.011(-0.680) 2,2

Beef 1956 1972 0.004***(2.861) 0.042***(3.503) -0.020***(-3.337) 2,1

Cocoa 1945 1955 -0.009**(-2.032) 0.053***(2.534) -0.005*(-1.681) 1,0

Coffee 1974 1987 0.005**(2.168) -0.018(-0.954) -0.011(-0.888) 1,0

Cotton 1929 1951 -0.031(-0.278) 0.002(0.025) -0.032(-0.990) 2,2

Hides 1919 1930 0.068***(4.448) 0.076**(2.41) -0.004**(-2.093) 0,1

Jute 1930 1965 -0.044(-0.421) 0.009(0.169) -0.040*(-1.856) 1,2

Maize 1918 1947 0.031(1.324) 0.012(10.29) -0.017***(-3.956) 0,3

Rice 1915 1980 -0.066(-0.498) -0.008(-1.456) -0.030**(-2.140) 2,2

Rubber 1911 1924 0.037(0.238) -0.971***(-11.234) -0.027***(-4.017) 2,0

Silver 1972 1983 -0.002(-1.074) 0.098***(4.725) -0.002(-0.260) 0,3

Sugar 1923 1936 0.035(1.303) -0.060(-1.058) -0.010**(-2.068) 0,1

Tea 1919 1966 0.018(0.275) 0.017(1.069) -0.024***(-2.646) 2,2

Timber 1966 1979 0.006***(3.839) 0.012(1.045) 0.002(0.361) 1,1

Tin 1972 1984 0.003(1.055) 0.010(0.594) -0.012(-1.031) 1,0

Tobacco 1915 1988 0.035***(2.713) 0.006***(4.388) -0.026***(-3.221) 2,2

Wheat 1919 1945 0.032**(2.246) -0.040***(-4.876) -0.168***(-7.402) 2,2

Wool 1913 1956 0.021(0.553) -0.014**(-2.302) -0.030***(-6.249) 0,1

Zinc 1917 1988 0.038**(1.995) 0.003(1.394) 0.027*(1.703) 0,1

Banana 1966 N.A. -0.002(-0.207) -0.008(-1.006) N.A. 1,2

Lead 1978 N.A. 0.012(0.218) -0.0299(-0.252) N.A. 2,2

Panel B. Estimated difference stationary models with breaks

Copper 1915 1925 0.041(0.727) -0.136**(-2.008) 0.018(0.821) 0,0

Lamb 1936 1956 0.003(1.475) -0.018*(-1.819) 0.016***(3.133) 0,3

Palm oil 1917 1927 -0.007(-0.418) -0.260***(-3.629) -0.006(-0.478) 2,2

Notes: and denote the first and second break dates respectively. The slope coefficients are reported for regime 1, 2 and 3. ***, ** and * denote significant at the 1%, 5% and 10% levels respectively. The final column represents the ARMA(p,q) specification. The numbers in parentheses denote the -rations.

45

Table 2.6 Relative Measures of a Prevalence of a Trend

(-) (+) (.)

Aluminum 0.000 0.000 1.000

Banana 0.000 0.000 1.000

Beef 0.324 0.676 0.000

Cocoa 0.907 0.093 0.000

Coffee 0.000 0.694 0.306

Copper 0.093 0.000 0.907

Cotton 0.000 0.000 1.000

Hides 0.713 0.287 0.000

Jute 0.389 0.000 0.611

Lamb 0.185 0.472 0.343

Lead 0.000 0.000 1.000

Maize 0.556 0.000 0.444

Palm oil 0.093 0.000 0.907

Rice 0.250 0.000 0.750

Rubber 0.889 0.000 0.111

Silver 0.000 0.102 0.898

Sugar 0.657 0.000 0.343

Tea 0.380 0.000 0.620

Timber 0.000 0.620 0.380

Tin 0.000 0.000 1.000

Tobacco 0.176 0.824 0.000

Wheat 0.815 0.185 0.000

Wool 0.870 0.000 0.130

Zinc 0.000 0.343 0.657

46

Figure 2.1 Relative Primary Commodity Prices

- Aluminum

1900 1920 1940 1960 1980 20000

1

2

3

4

5

6

7

- Banana

1900 1920 1940 1960 1980 20000.75

1.00

1.25

1.50

1.75

2.00

2.25

- Beef

1900 1920 1940 1960 1980 20000.00

0.25

0.50

0.75

1.00

1.25

1.50

- Cocoa

1900 1920 1940 1960 1980 20000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

- Coffee

1900 1920 1940 1960 1980 20000.00

0.25

0.50

0.75

1.00

1.25

1.50

- Copper

1900 1920 1940 1960 1980 20000.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

- Cotton

1900 1920 1940 1960 1980 20000.0

0.5

1.0

1.5

2.0

2.5

- Hides

1900 1920 1940 1960 1980 20000.5

1.0

1.5

2.0

2.5

3.0

- Jute

1900 1920 1940 1960 1980 20000.0

0.5

1.0

1.5

2.0

2.5

3.0

- Lamb

1900 1920 1940 1960 1980 20000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

- Lead

1900 1920 1940 1960 1980 20000.25

0.50

0.75

1.00

1.25

1.50

1.75

- Maize

1900 1920 1940 1960 1980 20000.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

47

Figure 2.1 Relative Primary Commodity Prices (continued)

- Palm oil

1900 1920 1940 1960 1980 20000.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

- Rice

1900 1920 1940 1960 1980 20000.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

- Rubber

1900 1920 1940 1960 1980 20000

2

4

6

8

10

12

14

- Silver

1900 1920 1940 1960 1980 20000.0

0.5

1.0

1.5

2.0

2.5

- Sugar

1900 1920 1940 1960 1980 20000

1

2

3

4

5

- Tea

1900 1920 1940 1960 1980 20000.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

- Timber

1900 1920 1940 1960 1980 20000.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

- Tin

1900 1920 1940 1960 1980 20000.0

0.2

0.4

0.6

0.8

1.0

1.2

- Tobacco

1900 1920 1940 1960 1980 20000.2

0.4

0.6

0.8

1.0

1.2

1.4

- Wheat

1900 1920 1940 1960 1980 20000.5

1.0

1.5

2.0

2.5

3.0

- Wool

1900 1920 1940 1960 1980 20000.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

- Zinc

1900 1920 1940 1960 1980 20000.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

48

CHAPTER 3

MORE POWERFUL FOURIER LM UNIT ROOT TESTS WITH NON-NORMAL ERRORS

The issue of whether an economic time series is best characterized by either a unit root or

a stationary process has assumed great importance in both the theoretical and the applied time

series econometrics literature. As a consequence, tests of the null hypothesis that a series is

integrated of order one, I(1), against the alternative hypothesis that it is integrated of order zero,

I(0), have received much attention. Now it has been well documented that traditional unit root

tests tend to have low power problem when the deterministic components of the testing time

series are mis-specified. In a seminal paper, Perron (1989) first documented that failure to

account for a structural break can result in the standard Dickey-Fuller (DF) test lose power

significantly. To provide a remedy, Perron suggested a modified DF unit root test that includes

dummy variables to control for known structural breaks in the level and slopes. Perron’s paper

has three assumptions that are hardly satisfied in empirical researches, which includes:(i) the

structural breaks are assumed to be known a priori, (ii) the structural breaks are assumed to be

steep or abrupt, and (iii) the error term is assumed to follow a standard normal distribution,

respectively. Subsequent papers further modified unit root tests have been developed to relax one

or more of these assumptions.

In empirical study, the information on breaks is generally unknown. Various tests are

developed to allow for one or more unknown structural breaks that are determined endogenously

49

from the data. The most popular endogenous break unit root tests include the tests developed in

Zivot and Andrews (1992), Lumsdaine and Papell (1997), Perron (1997), Vogelsang and Perron

(1998), and Lee and Strazicich (2003, 2004). These tests are featured with searching for possible

structural breaks using a grid search, which the breakpoint is determined where the t-statistic on

the unit root hypothesis is minimized or the t-statistic/f-statistic for the break coefficient is

maximized. However, most of these tests are subject to the so-called spurious rejection problem

when they assume no break under the null; see Lee and Strazicich (2001). Lee and Strazicich

(2003) allow for breaks both under the null and alternative hypotheses, and provide a remedy for

this problem. Lee, Strazicich and Meng (2012) compared the performance of these endogenous

break unit root tests, and they found that all the above endogenous tests have some drawbacks,

some of which are serious and others are minor. In a following paper, Lee, Meng and Strazicich

(2012) suggested a test based on a two-step procedure, which searches for a structural break in

the first step and test for the unit root in the second step. The Monte Carlo experiments show the

two-step test does not exhibit size distortions, has power similar to that of the exogenous tests,

and accurately identifies the breaks.

However, there are several minor problems for the two-step unit root tests. Firstly, the

maximum number of structural breaks need to be specified in advance; secondly, the searching

procedure works poorly if breaks are located at the beginning or the end of the testing series;

lastly, the searching procedure works poorly if breaks are adjacent to each other.

Typically structural breaks in the deterministic terms are assumed to be steep and abrupt,

yet a number of researchers argue that the structural changes can be gradual and smooth. To

allow for such structural changes, Leybourne et al. (1998) and Kapetanios et al. (2003) develop

unit-root tests such that the deterministic component of the series is a smooth transition process.

50

Similar to the exogenous break unit root test of Perron (1989), the information about the break

date and the specific functional form of the nonlinearity needs to be specified as a priori. Another

problem of this type of nonlinear unit root tests is the spurious rejection problem. Meng (2012)

provide evidence that assuming no break under the null in these nonlinear unit root tests causes

the test statistic to diverge and lead to significant rejections of the unit root null when the true

data-generating process (DGP) is a unit root with nonlinear deterministic term(s). A third

problem is although it is possible to allow for more smooth breaks, these tests are not powerful

since too many parameters need to be estimated.

Recently, Enders and Lee (2012a, hereafter EL) suggest a different approach to address

structural breaks. They suggest a LM based unit root test which uses a variant of Gallant’s (1981)

flexible Fourier form (hereafter FLM) to control for the unknown nature of the structural

break(s).1 They follow Becker et al. (2004, 2006) and illustrate that the essential characteristic

of a series with one or more structural breaks can often be captured using a small number of low-

frequency components from a Fourier approximation. First of all, their tests allow for structural

break under both the null hypothesis and the alternative hypothesis, thus they are free of the

spurious rejection problem. Secondly, these tests use Fourier approximation to circumvent the

needs to assume the number of breaks, the break date, and the specific form of the break(s).

Their Monte Carlo simulations prove their test has good size and power properties for the

various types of break forms often used in empirical economic analysis.

My essay 1 develops a Residual Augmented Least Squares Lagrange Multiplier (RALS-

LM) test which utilizes the information that exists when the errors in the testing equation exhibit

1 Similar Dickey-Fuller Generalized Least Squares (DF-GLS) based Fourier unit root test and DF based Fourier unit root test are provided in Rodrigues and Taylor (2012) and Enders and Lee (2012b).

51

any departure from normality, such as non-linearity, asymmetry, or fat-tailed distributions.2 The

underlying idea of the RALS procedure is appealing because it is intuitive and easy to implement.

If the errors are non-normal, the higher moments of the residuals contain the information on the

nature of the non-normality. The RALS procedure conveniently utilizes these moments in a

linear testing equation without the need for a priori information on the nature of the non-

normality, such as the density function or the precise functional form of any non-linearity. The

simulation result showsthe power gain of the RALS-LM tests over the usual LM tests is

considerable when the error term is asymmetric or has a fat-tailed distribution.

Non-normality is common to find when dealing with real-world data (Im, Lee and

Tieslau, 2012).In many cases, however, it is hard to distinguish between non-normal

distributions and some forms of non-linearity. In addition, certain economic time series might

contain a mixture of both non-linearity and non-normality. If a specific nonlinear form is known,

it would be proper to utilize non-linear unit root tests with the specific form;3 if the true density

of the non-normal error is known, it is also possible to use maximum likelihood estimators (MLE)

to get a more efficient test. Unfortunately, the information about the non-linear forms and non-

normal errors is generally unknown a priori. In this regards, we explore a joint treatment of both

unknown forms of structural breaks and unknown forms of non-normal errors by extending the

FLM unit root tests of EL using the RALS estimation procedure of Im and Schmidt (2008). We

refer to this test as RALS-FLM unit root tests. The FLM type of tests could be used to control

for smooth structural breaks of an unknown functional form and the RALS procedure could

2 Im, Lee and Tieslau (2012) suggests a similar test based on the traditional DF unit root tests; Im, Lee and Lee (2012) suggests applying the RALS procedure to the cointegration tests. Both papers confirm power gain if the error term follows a non-normal distribution. 3 There is another problem for those nonlinear unit root tests. Lee, Meng and Lee (2011) found the presence of non-normal errors in nonlinear unit root tests will lead to a significant loss of power, although non-normal errors do not pose a problem in the usual unit root tests since the least square estimator will still be the most efficient under certain ideal conditions regardless of normal or non-normal errors.

52

utilize information contained in non-normal errors. With our new proposed tests, it is possible to

test for a unit root without having to model the precise form of the structural break, and it is also

possible to improve the testing power when the error term follows a non-normal distribution

without the need to specify a clear form of the non-normality. Our simulation results confirm

significant power gains over the FLM tests.

This essay proceeds as follows. In Section 3.1, we discuss the FLM Tests and propose the

RALS-FLM tests. In Section 3.2, we provide simulation results to examine the size and power

properties and compare them with those of the FLM tests. Section 3.3 provides concluding

remarks.

3.1. The RALS-FLM Test

To begin with our analysis, we consider data generated according to the following DGP:

, , (3.1.1)

, (3.1.2)

where we the initial condition, , to be an random variable. We are interested in testing

the null hypothesis of a unit root ( ) against the alternative hypothesis of a stationary

( ) in equation (3.1.2).If the functional form of is known, it is possible to test the

unit root directly. If the form of is unknown, we follow EL and approximate using the

Fourier expansion:

EL point out that the deterministic components in the above equation could be used to

approximate nonlinearity for any desired level of accuracy. However, as a practice matter, EL

recommends using a small number of frequencies to avoid losing too many degrees of freedom

53

and an overfitting problem. In light of the observation made in EL and Becker et al. (2006) that

a single Fourier frequency generally approximates nonlinear structural break well, for the time

being, we consider only a single frequency k. This enables our DGP to be rewritten as follows:

,

. (3.3.3)

Following the LM (score) principle, we impose the null restriction and consider the

following regression in first-differences:

. (3.1.4)

The estimated coefficients are denote as , , and , respectively. The unit root test statistics

are then obtained from the following regression:

. (3.1.5)

Here, denotes the LM detrended series, which defined as:

, (3.1.6)

where , and is the first observation of .

Subtracting in (3.1.6) makes the initial value and the ending value of the detrended series to be

zero with and . We let denote the t-statistic for the null hypothesis

in (3.1.5). The asymptotic distribution of was shown in EL Theorem 1.

We next explain how to utilize the information in non-normal errors to improve the

testing power for the FLM test. We adopt the RALS estimation procedure as suggested in Im

and Schmidt (2008). The RALS procedure is to augment the following term to the Fourier-

LM testing regression (3.1.5):

, (3.1.7)

54

where denote the residuals from the FLM regression (3.1.5), ,

, and . Specifically, we let , which

involves the second and third moments of the residual . Then, letting , the

augmented term can be given as:

. (3.1.8)

The first term in is associated with the moment condition , which is the

condition of no heteroskedasticity. This condition improves the efficiency of the estimator of

when the error terms are not symmetric. The second term in improves efficiency unless

. In general, knowledge of higher moments are uninformative if

. This is the redundancy condition. The normal distribution is the only

distribution that satisfies the redundancy condition. However, if the distribution of the error term

is not normal, the condition is not satisfied. In such cases, one may increase efficiency by

augmenting the testing regression (3.1.5) with . Then the RALS based FLM test statistic is

obtained from the regression:

. (3.1.9)

We denote this test as RALS-FLM test and the corresponding -statistic for from

regression (3.1.9) as . Then it can be shown that the asymptotic distribution of

is given as follows:

, (3.1.10)

where denotes the FLM distribution as defined in (3.1.5), indicates the standard normal

distribution, and reflects the relative ratio of the variances of two error terms:

, (3.1.11)

and are the error term in regression (3.1.5) and (3.1.9), respectively.

55

The critical values for the null hypothesis of a unit root of the single frequency FLM test

in EL depend on the frequency , however, our RALS-FLM test statistic depend on both the

frequency and the correlation . From equation (3.1.11) we can see that could takes any value

between 0 and 1. In particular, if = 1, which is the case when there is no extra information in

the second and third moment of the error term, the distribution for the RALS-FLM statistic is the

same as the distribution of FLM statistic. If = 0, which is the case when the two error terms are

orthogonal to each other, the distribution for RALS-FLM statistic is the same as the standard

normal distribution. If takes any value between 0 and 1, the RALS-FLM statistic distribution is

a weighted mix of the FLM statistic distribution and a standard normal distribution. Since the

critical values for the RALS-FLM test under the null hypothesis do not depend on the

coefficients of the Fourier terms or other deterministic terms, we can tabulate critical values

using simulations. The critical values for single Fourier frequency RALS-FLM test are reported

in Table 3.1.a.

In practice, the true value of is not available, we could estimate it using the residuals

from regression (3.1.5) and (3.1.9):

,

where is the usual estimate of the error variance from FLM regression (3.1.5), and is the

estimate of the error variance in the RALS-FLM regression (3.1.9).

We next consider the RALS-FLM test with unknown deterministic trend. In this case,

two issues need to be taken care of. The first issue is pretesting for a nonlinear trend. If

nonlinear trend is absent from the DGP, the RALS-LM test suggested in my Essay 1will be more

powerful than our RALS-FLM test. Following EL, a F test is applied to test the null hypothesis

of linearity against the alternative hypothesis of a Fourier trend with given frequency . EL

56

denote this test as and the critical values are reported in Panel b of Table 1 in EL. The

second issue is selecting the frequency that provides the best fit to the unknown nonlinear trend.

Again, we can apply the data-driven grid search method suggested in EL that select the

frequency minimizes the sum of squared residuals from equation (3.1.5). Logically, we need to

pretest for a nonlinear trend before we model the nonlinear trend using a Fourier expansion with

a frequency . In empirical study, we need to do this reversely. To summarize, the data-driven

RALS-FLM testing procedure with unknown deterministic trend can be summarized as follows:

Step 1: Estimate (3.1.5) for all integer values of such that .4 For each

integer value of , we simultaneously determine the optimal lags using a general to specific

method suggested by Ng and Perron (1995).5 The regression with the smallest sum of squared

residuals (SSR) yields and the corresponding .The resulting test is denoted as .

Step 2: For the optimal and identified in the first step, we test for the existence of a

nonlinear trend. For this, we perform the usual-test for the null hypothesis of linearity (i.e.

in equation (3.1.3)), and compare the -statistic with the critical value for . If

the sample is sufficiently large, we can employ the RALS-FLM test with frequency and

number of lags . If the null is not rejected, then we can employ the RALS-LM unit root

proposed by Essay 1 without a nonlinear trend.

The critical values of are reported in Table 3.1.b.

4 EL suggest choosing from the integer values 1 through 5. However, this imposes a potentially important restriction on the trigonometric component of the trend function in that it must start and end at the same position relative to the linear trend component of the DGP. To avoid this restriction, it is also possible to consider frictional frequencies. 5 For each frequency , we set the maximum lags to be and augment the regression with lags. If the -statistic for the th lag is significant, we set the optimal lag for frequency equal to . If not, we continue with setting the number of lags equal to . The searching procedure continues until the number of lags equal to zero or the -statistic for the th lag is significant.

57

We also consider the RALS-FLM tests which allow for cumulative Fourier frequencies.

The motivation of using cumulative frequencies is that they can provide a more precise

approximation. We denote as the number of frequencies used in the test. If = 1, the test is

the same as the single frequency test using = 1. If , all the frequencies from 1 to are

used in the regression. Since the individual frequency components are orthogonal to each other,

it can be easily shown that the critical values depend only on and . We denote the RALS-

FLM test with cumulative frequencies as to distinguish it from the test using a

single frequency. The appropriate critical values for for = 2, 3, = 0, 0.1,…,

1.0, and = 100, 200, and 2500 are reported in Table 3.2. Although using cumulative

frequencies can offer more precise estimate, there is a caveat that they can yield significant loss

of power due to over-fitting of the data.

3.2. The Monte Carlo Experiments

In this section, we investigate the performance of the suggested RALS-FLM test using

Monte Carlo simulation experiments. All simulations are performed using 20, 000 replications

with WinRATS 7.2.The DGP was given in equation (3.1.3) with the initial value follows a

N�0,1� distribution. To examine the impact of the distribution of the error term on the

performance of RALS-FLM test, we allow the error term to follow seven types of non-normal

distribution: distribution with , and -distribution with . We also

allow the error term following a standard normal distribution for comparison purpose.To

conserve space, we report only results using 5% nominal critical values with sample size equal to

100 and 200.6

6 The results for = 500 and 2500 are available from the author.

58

To begin with, we show that the test will depend on , but will be invariant

to the magnitude of the and . To do this, we assume that the value of is known a priori,

and we change the combinations of and in the DGP. The results in Table 3.3 show that the

size of our tests is close to 5% in all cases with different values of , and , regardless of the

distribution of the error term. The test shows a mild over rejection when the error

term follows a standard normal distribution. This is expected because the RALS procedure

barely gets any extra information when the error term follows a normal distribution; nevertheless,

the test still augments the higher moment condition terms, which cause a loss in the

testing efficiency. When the sample size is small, the power of is low. The

tests show significantly improved power over the tests when the errors follow non-normal

distribution with either a - or -distribution. The power gain of the test is greater

when the degrees of freedom of the or the -distribution are smaller, implying more

asymmetric patterns or fatter tails of the error distribution. The power gain falls when the

distribution of the error term towards a normal distribution. When the error term follows a

normal distribution, the adjusted power of is slightly lower than the adjust power of

test. The power for the test improves as gets bigger; this finding is consistent

with the finding in EL for the tests. When sample size increase, the size distortion

becomes smaller and the power increase greatly, implying that the test is consistent. When

sample size is greater than 500, the testing power for both tests exceeds 99% when = .9.

Next, we examine the cases where is treated as unknown and estimated from the data.

We follow the two-step procedure as discussed in section two. We test the null hypothesis that

the trend in the DGP is linear by using the critical values of reported in Table 3.1.b. If the

null hypothesis of a linear trend is rejected, we apply the test; otherwise, we apply

59

the test. The simulation results are reported in Table 3.4. Similar to what we find in Table

3.3, both tests display correct size for different combinations of , , and . While the power

of tests is robust to different distributions in the error term, tests are more

powerful when the error term follows a non-normal distribution. To our surprise, tests using

estimated value of are more powerful than tests using known value of when , and both

equal to zero and the error term follows a severe non-normal distribution.

The results for tests using cumulative frequencies are reported in Table 3.5. The DGP is

specified as equation (3.1.1 and 3.1.2). Here we choose = = 0 for = 1, 2, and 3, but the

results would be invariant by using other values of and . The results indicate that the

test shows a mild size distortion, which becomes serious as becomes larger. While

the test still shows a significant power gain when the error term follows a non-

normal distribution compared with test, the power for both tests diminishes quickly as

additional frequencies are added to the estimation equation. For example, when sample size is

100 and the error term follows a distribution, the power for with = 1 is 0.896,

and the power drops to 0.663 if = 2, and 0.375 if = 3.As such, we suggest researchers use a

small value of if cumulative frequencies are needed in the estimation. In a former paper,

Bierens (1997) developed a unit root test using a linear function of Chebishev polynomials to

approximate the function of time. Contrary to our finding, Bierens found the testing power of his

test appears to increase as increases. We attribute this power increase to spurious rejection

problem. In his test, Bierens assumes no nonlinearity under the null hypothesis. Thus rejection

of the null hypothesis doesn’t necessarily mean that the testing series is stationary, because there

is still possibility that the series is a unit root process with nonlinear trend. Our tests allow for

nonlinear trend both under the null and alternative hypotheses, and are free of this problem.

60

3.3. Concluding Remarks

In this essay, we develops a RALS based Fourier-LM test. While the Fourier-LM type of

tests can be used to control for smooth breaks or multiple breaks of an unknown functional form,

the RALS procedure can utilize additional information contained in non-normal errors. One

important feature of the new test is that the knowledge of the underlying type of non-normal

distribution or the precise functional form of the structural breaks is not required. By using

parsimonious number of parameters to control for possible breaks in the deterministic term and

use the additional information lied in nonlinear moments of the error term, the power of RALS-

Fourier-LM test increase dramatically when the error term follows a non-normal distribution.

61

3.4. Appendix

Appendix Table 3.1 Critical Values of Transformed RALS-LM Test with Trend Break

%

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

50

1

1 -2.333 -3.101 -3.390 -3.590 -3.745 -3.878 -3.996 -4.108 -4.222 -4.324 -4.421

5 -1.640 -2.432 -2.736 -2.955 -3.131 -3.274 -3.404 -3.514 -3.613 -3.698 -3.790

10 -1.286 -2.090 -2.396 -2.619 -2.799 -2.958 -3.091 -3.207 -3.309 -3.405 -3.487

2

1 -2.333 -3.283 -3.639 -3.896 -4.113 -4.301 -4.459 -4.606 -4.752 -4.883 -5.004

5 -1.640 -2.619 -2.995 -3.270 -3.494 -3.682 -3.848 -3.999 -4.133 -4.259 -4.377

10 -1.286 -2.268 -2.651 -2.933 -3.162 -3.358 -3.532 -3.686 -3.824 -3.952 -4.074

100

1

1 -2.322 -3.086 -3.365 -3.564 -3.713 -3.840 -3.948 -4.046 -4.127 -4.207 -4.282

5 -1.646 -2.427 -2.716 -2.923 -3.090 -3.233 -3.352 -3.458 -3.554 -3.639 -3.712

10 -1.283 -2.078 -2.377 -2.598 -2.775 -2.924 -3.051 -3.165 -3.262 -3.350 -3.434

2

1 -2.322 -3.264 -3.597 -3.843 -4.046 -4.212 -4.371 -4.498 -4.610 -4.717 -4.804

5 -1.646 -2.604 -2.965 -3.229 -3.443 -3.622 -3.778 -3.918 -4.040 -4.146 -4.252

10 -1.283 -2.247 -2.622 -2.899 -3.121 -3.309 -3.473 -3.621 -3.748 -3.865 -3.971

300

1

1 -2.332 -3.079 -3.337 -3.525 -3.672 -3.788 -3.894 -3.977 -4.064 -4.128 -4.188

5 -1.642 -2.419 -2.705 -2.911 -3.077 -3.213 -3.327 -3.430 -3.518 -3.600 -3.669

10 -1.276 -2.066 -2.369 -2.588 -2.766 -2.915 -3.039 -3.144 -3.240 -3.326 -3.400

2

1 -2.332 -3.245 -3.582 -3.822 -4.007 -4.158 -4.288 -4.405 -4.513 -4.606 -4.683

5 -1.642 -2.587 -2.941 -3.200 -3.405 -3.579 -3.730 -3.861 -3.977 -4.080 -4.173

10 -1.276 -2.234 -2.604 -2.873 -3.091 -3.282 -3.440 -3.580 -3.703 -3.816 -3.913

1000

1

1 -2.345 -3.102 -3.368 -3.548 -3.688 -3.798 -3.894 -3.983 -4.049 -4.114 -4.175

5 -1.652 -2.433 -2.725 -2.927 -3.091 -3.222 -3.333 -3.429 -3.516 -3.594 -3.654

10 -1.291 -2.080 -2.379 -2.599 -2.771 -2.919 -3.046 -3.153 -3.245 -3.325 -3.395

2

1 -2.345 -3.258 -3.586 -3.821 -4.006 -4.153 -4.281 -4.398 -4.496 -4.592 -4.672

5 -1.652 -2.599 -2.953 -3.212 -3.419 -3.591 -3.741 -3.867 -3.982 -4.079 -4.158

10 -1.291 -2.242 -2.610 -2.881 -3.100 -3.286 -3.444 -3.579 -3.703 -3.811 -3.907

Notes: denotes the sample size; denotes the break number; denotes the coefficient in equation (12). The transformation does not influence the critical value, so this table can be used on non-transformed RALS-LM test when the structural breaks are evenly distributed. When = 0, the critical values are the same with those of the standard normal distribution; when = 1, the critical values are the same with transformed LM test or non-transformed LM test with structural breaks evenly distributed in the data.

62

Appendix Table 3.2 Size and Power Property ( 100, 1)

(1) (2) (3) (4) (2) (3) (4) N (0,1)

Size Property

1

.25

0.064 0.072 0.075 0.071 0.051 0.053 0.059 0.087

0.040 0.050 0.044 0.047 0.033 0.037 0.040 0.072

0.041 0.043 0.044 0.045 0.043 0.050 0.047 0.047

0.030 0.033 0.032 0.035 0.029 0.034 0.036 0.040

0.036 0.033 0.031 0.029 0.046 0.038 0.032 0.036

.50

0.045 0.050 0.058 0.063 0.042 0.048 0.057 0.092

0.044 0.044 0.045 0.048 0.037 0.045 0.050 0.051

0.049 0.052 0.050 0.053 0.060 0.057 0.058 0.051

Power Property

0.8

.25

0.975 0.954 0.923 0.881 0.771 0.616 0.530 0.442

0.988 0.973 0.954 0.927 0.832 0.714 0.631 0.518

0.363 0.365 0.365 0.377 0.366 0.365 0.371 0.370

0.462 0.458 0.460 0.459 0.450 0.452 0.455 0.451

0.341 0.342 0.345 0.352 0.319 0.334 0.342 0.335

.50

0.991 0.978 0.961 0.936 0.853 0.723 0.655 0.546

0.487 0.476 0.485 0.489 0.478 0.470 0.482 0.478

0.364 0.362 0.372 0.380 0.339 0.364 0.373 0.362

.9

.25

0.880 0.771 0.679 0.600 0.463 0.293 0.224 0.205

0.913 0.802 0.692 0.598 0.460 0.281 0.219 0.197

0.112 0.120 0.123 0.129 0.119 0.127 0.128 0.138

0.106 0.126 0.126 0.124 0.113 0.122 0.123 0.133

0.087 0.089 0.089 0.093 0.097 0.095 0.091 0.096

.5

0.909 0.811 0.697 0.613 0.474 0.306 0.244 0.230

0.131 0.140 0.142 0.140 0.130 0.143 0.141 0.154

0.109 0.114 0.116 0.111 0.119 0.122 0.111 0.123

Notes: denotes the coefficient for the DGP; denotes the break location which defined as , where is the break location; denotes the test statistics. , ,

denote the test statistic for RALS-LM test, LM test and DF test, respectively; means transformed test. When 0.5, the size and power for the transformed tests and untransformed tests are the same, we report them together to save space.

63

Table 3.1.a Critical Values of Tests

%

.0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.

100 1 1 -2.332 -3.209 -3.529 -3.764 -3.955 -4.11 -4.246 -4.373 -4.497 -4.605 -4.692

5 -1.652 -2.544 -2.886 -3.134 -3.335 -3.503 -3.652 -3.782 -3.902 -4.008 -4.112

10 -1.287 -2.189 -2.539 -2.793 -3.003 -3.183 -3.34 -3.48 -3.605 -3.717 -3.822

2 1 -2.332 -2.974 -3.233 -3.413 -3.567 -3.698 -3.819 -3.935 -4.032 -4.135 -4.221

5 -1.652 -2.307 -2.561 -2.749 -2.905 -3.038 -3.16 -3.271 -3.377 -3.48 -3.573

10 -1.287 -1.953 -2.211 -2.4 -2.557 -2.694 -2.814 -2.926 -3.034 -3.134 -3.224

3 1 -2.332 -2.923 -3.138 -3.286 -3.408 -3.518 -3.618 -3.713 -3.799 -3.887 -3.959

5 -1.652 -2.251 -2.475 -2.639 -2.771 -2.88 -2.974 -3.067 -3.145 -3.22 -3.303

10 -1.287 -1.899 -2.13 -2.298 -2.431 -2.543 -2.642 -2.73 -2.812 -2.889 -2.961

200 1 1 -2.307 -3.191 -3.513 -3.748 -3.928 -4.078 -4.206 -4.325 -4.431 -4.527 -4.623

5 -1.644 -2.527 -2.87 -3.115 -3.315 -3.487 -3.634 -3.759 -3.878 -3.978 -4.077

10 -1.283 -2.182 -2.529 -2.783 -2.992 -3.167 -3.323 -3.462 -3.583 -3.699 -3.798

2 1 -2.307 -2.953 -3.201 -3.38 -3.53 -3.663 -3.781 -3.896 -3.996 -4.098 -4.195

5 -1.644 -2.304 -2.559 -2.748 -2.904 -3.04 -3.16 -3.273 -3.374 -3.472 -3.56

10 -1.283 -1.948 -2.208 -2.403 -2.56 -2.697 -2.819 -2.93 -3.037 -3.132 -3.22

3 1 -2.307 -2.896 -3.119 -3.28 -3.404 -3.512 -3.61 -3.708 -3.802 -3.888 -3.973

5 -1.644 -2.251 -2.479 -2.64 -2.774 -2.885 -2.984 -3.075 -3.155 -3.232 -3.3

10 -1.283 -1.897 -2.131 -2.303 -2.441 -2.558 -2.659 -2.748 -2.827 -2.901 -2.974

2500 1 1 -2.32 -3.185 -3.493 -3.727 -3.905 -4.051 -4.183 -4.29 -4.392 -4.48 -4.541

5 -1.641 -2.52 -2.854 -3.096 -3.295 -3.461 -3.609 -3.737 -3.846 -3.946 -4.032

10 -1.278 -2.175 -2.523 -2.779 -2.987 -3.162 -3.315 -3.446 -3.565 -3.675 -3.77

2 1 -2.32 -2.974 -3.215 -3.4 -3.547 -3.672 -3.782 -3.886 -3.974 -4.055 -4.141

5 -1.641 -2.309 -2.557 -2.741 -2.895 -3.026 -3.147 -3.257 -3.353 -3.452 -3.542

10 -1.278 -1.946 -2.205 -2.396 -2.554 -2.692 -2.814 -2.924 -3.027 -3.125 -3.218

3 1 -2.32 -2.918 -3.127 -3.279 -3.399 -3.515 -3.623 -3.719 -3.801 -3.867 -3.935

5 -1.641 -2.26 -2.484 -2.646 -2.778 -2.89 -2.985 -3.07 -3.154 -3.231 -3.305

10 -1.278 -1.897 -2.133 -2.303 -2.441 -2.56 -2.663 -2.754 -2.834 -2.909 -2.987

Note: To save space, we only report the critical values for = 100, 200, and 2500, and = 1, 2, and 3.

64

Table 3.1.b Critical Values of

T = 100 T = 200 T = 500 T = 2500

1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10%

6.940 5.037 4.227 6.458 4.808 4.069 6.399 4.646 3.907 6.194 4.615 3.891

Table 3.2 Critical Values of

% .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.

100 2 1 -2.332 -3.456 -3.887 -4.193 -4.449 -4.669 -4.866 -5.035 -5.188 -5.334 -5.498

5 -1.652 -2.784 -3.231 -3.559 -3.826 -4.056 -4.26 -4.444 -4.611 -4.763 -4.907

10 -1.287 -2.43 -2.882 -3.216 -3.493 -3.73 -3.941 -4.13 -4.303 -4.463 -4.617

3 1 -2.332 -3.663 -4.176 -4.556 -4.866 -5.139 -5.383 -5.609 -5.818 -6.001 -6.181

5 -1.652 -2.99 -3.521 -3.912 -4.244 -4.527 -4.777 -5.002 -5.209 -5.401 -5.585

10 -1.287 -2.642 -3.177 -3.576 -3.909 -4.201 -4.457 -4.695 -4.907 -5.103 -5.286

200 2 1 -2.307 -3.416 -3.832 -4.137 -4.384 -4.598 -4.783 -4.952 -5.109 -5.249 -5.391

5 -1.644 -2.769 -3.209 -3.538 -3.802 -4.025 -4.221 -4.397 -4.56 -4.705 -4.839

10 -1.283 -2.422 -2.868 -3.201 -3.474 -3.71 -3.92 -4.105 -4.271 -4.424 -4.563

3 1 -2.307 -3.616 -4.129 -4.503 -4.807 -5.075 -5.301 -5.506 -5.69 -5.869 -6.037

5 -1.644 -2.967 -3.492 -3.879 -4.201 -4.475 -4.719 -4.939 -5.137 -5.321 -5.483

10 -1.283 -2.619 -3.148 -3.548 -3.875 -4.159 -4.407 -4.634 -4.839 -5.032 -5.207

2500 2 1 -2.32 -3.422 -3.841 -4.138 -4.371 -4.576 -4.755 -4.906 -5.047 -5.162 -5.271

5 -1.641 -2.76 -3.19 -3.51 -3.766 -3.985 -4.181 -4.352 -4.511 -4.648 -4.774

10 -1.278 -2.408 -2.85 -3.179 -3.447 -3.678 -3.882 -4.064 -4.226 -4.373 -4.508

3 1 -2.32 -3.614 -4.109 -4.463 -4.759 -5.008 -5.226 -5.421 -5.6 -5.747 -5.88

5 -1.641 -2.957 -3.466 -3.85 -4.16 -4.425 -4.663 -4.869 -5.059 -5.229 -5.388

10 -1.278 -2.602 -3.128 -3.519 -3.84 -4.115 -4.361 -4.58 -4.783 -4.963 -5.131

Note: Critical values for = 1 are omitted because they are the same with test using single frequency with = 1.

65

Table 3.3 Finite Sample Performance with Known Frequencies (T = 100)

DGP TESTS

Distribution of Error ��0,1�

1 0 0 1.0 0.036 0.047 0.059 0.065 0.042 0.049 0.053 0.100

0.041 0.043 0.045 0.047 0.038 0.045 0.047 0.051

0.9(A) 0.910 0.757 0.596 0.473 0.428 0.221 0.169 0.106

0.125 0.121 0.118 0.120 0.120 0.119 0.119 0.113

0 5 1.0 0.036 0.047 0.059 0.065 0.042 0.049 0.053 0.100

0.041 0.043 0.045 0.047 0.038 0.045 0.047 0.051

0.9(A) 0.910 0.757 0.596 0.473 0.428 0.221 0.169 0.106

0.125 0.121 0.118 0.120 0.120 0.119 0.119 0.113

3 0 1.0 0.036 0.047 0.059 0.065 0.042 0.049 0.053 0.100

0.041 0.043 0.045 0.047 0.038 0.045 0.047 0.051

0.9(A) 0.910 0.757 0.596 0.473 0.428 0.221 0.169 0.106

0.125 0.121 0.118 0.120 0.120 0.119 0.119 0.113

3 5 1.0 0.036 0.047 0.059 0.065 0.042 0.049 0.053 0.100

0.041 0.043 0.045 0.047 0.038 0.045 0.047 0.051

0.9(A) 0.910 0.757 0.596 0.473 0.428 0.221 0.169 0.106

0.125 0.121 0.118 0.120 0.120 0.119 0.119 0.113

2 0 0 1.0 0.044 0.051 0.059 0.063 0.043 0.044 0.047 0.082

0.045 0.045 0.047 0.046 0.040 0.045 0.047 0.050

0.9(A) 0.941 0.849 0.736 0.645 0.598 0.398 0.302 0.196

0.219 0.223 0.215 0.215 0.224 0.222 0.220 0.210

0 5 1.0 0.044 0.051 0.059 0.063 0.043 0.044 0.047 0.082

0.045 0.045 0.047 0.046 0.040 0.045 0.047 0.050

0.9(A) 0.941 0.849 0.736 0.645 0.598 0.398 0.302 0.196

0.219 0.223 0.215 0.215 0.224 0.222 0.220 0.210

3 0 1.0 0.044 0.051 0.059 0.063 0.043 0.044 0.047 0.082

0.045 0.045 0.047 0.046 0.040 0.045 0.047 0.050

0.9(A) 0.941 0.849 0.736 0.645 0.598 0.398 0.302 0.196

0.219 0.223 0.215 0.215 0.224 0.222 0.220 0.210

3 5 1.0 0.044 0.051 0.059 0.063 0.043 0.044 0.047 0.082

0.045 0.045 0.047 0.046 0.040 0.045 0.047 0.050

0.9(A) 0.941 0.849 0.736 0.645 0.598 0.398 0.302 0.196

0.219 0.223 0.215 0.215 0.224 0.222 0.220 0.210

Note: 0.9 (A) means size adjusted power.

66

Table 3.3 (Continued), = 200

DGP TESTS

Distribution of Error ��0,1�

1 0 0 1.0 0.042 0.051 0.051 0.055 0.042 0.042 0.046 0.069

0.042 0.048 0.046 0.046 0.039 0.043 0.048 0.050

0.9(A) 0.999 0.994 0.981 0.953 0.905 0.732 0.600 0.350

0.406 0.391 0.401 0.399 0.412 0.402 0.386 0.380

0 5 1.0 0.042 0.051 0.051 0.055 0.042 0.042 0.046 0.069

0.042 0.048 0.046 0.046 0.039 0.043 0.048 0.050

0.9(A) 0.999 0.994 0.981 0.953 0.905 0.732 0.600 0.350

0.406 0.391 0.401 0.399 0.412 0.402 0.386 0.380

3 0 1.0 0.042 0.051 0.051 0.055 0.042 0.042 0.046 0.069

0.042 0.048 0.046 0.046 0.039 0.043 0.048 0.050

0.9(A) 0.999 0.994 0.981 0.953 0.905 0.732 0.600 0.350

0.406 0.391 0.401 0.399 0.412 0.402 0.386 0.380

3 5 1.0 0.042 0.051 0.051 0.055 0.042 0.042 0.046 0.069

0.042 0.048 0.046 0.046 0.039 0.043 0.048 0.050

0.9(A) 0.999 0.994 0.981 0.953 0.905 0.732 0.600 0.350

0.406 0.391 0.401 0.399 0.412 0.402 0.386 0.380

2 0 0 1.0 0.045 0.053 0.051 0.057 0.041 0.043 0.043 0.063

0.045 0.048 0.048 0.049 0.040 0.046 0.047 0.050

0.9(A) 0.998 0.995 0.988 0.974 0.954 0.864 0.804 0.581

0.628 0.619 0.611 0.612 0.682 0.626 0.626 0.609

0 5 1.0 0.045 0.053 0.051 0.057 0.041 0.043 0.043 0.063

0.045 0.048 0.048 0.049 0.040 0.046 0.047 0.050

0.9(A) 0.998 0.995 0.988 0.974 0.954 0.864 0.804 0.581

0.628 0.619 0.611 0.612 0.682 0.626 0.626 0.609

3 0 1.0 0.045 0.053 0.051 0.057 0.041 0.043 0.043 0.063

0.045 0.048 0.048 0.049 0.040 0.046 0.047 0.050

0.9(A) 0.998 0.995 0.988 0.974 0.954 0.864 0.804 0.581

0.628 0.619 0.611 0.612 0.682 0.626 0.626 0.609

3 5 1.0 0.045 0.053 0.051 0.057 0.041 0.043 0.043 0.063

0.045 0.048 0.048 0.049 0.040 0.046 0.047 0.050

0.9(A) 0.998 0.995 0.988 0.974 0.954 0.864 0.804 0.581

0.628 0.619 0.611 0.612 0.682 0.626 0.626 0.609

67

Table 3.4 Finite Sample Performance of Tests using -test, T = 100

DGP TESTS

Distribution of Error ��0,1�

1 0 0 1.0 0.054 0.067 0.078 0.083 0.055 0.067 0.076 0.121

0.076 0.077 0.078 0.081 0.068 0.074 0.077 0.084

0.9(A) 0.969 0.887 0.747 0.613 0.559 0.263 0.151 0.072

0.064 0.068 0.064 0.065 0.070 0.069 0.071 0.064

0 5 1.0 0.111 0.091 0.083 0.077 0.058 0.048 0.054 0.091

0.044 0.047 0.049 0.046 0.058 0.044 0.048 0.050

0.9(A) 0.489 0.385 0.323 0.279 0.290 0.196 0.151 0.113

0.130 0.129 0.123 0.130 0.071 0.127 0.129 0.124

3 0 1.0 0.139 0.105 0.091 0.086 0.054 0.054 0.060 0.098

0.055 0.056 0.059 0.060 0.065 0.055 0.060 0.062

0.9(A) 0.509 0.393 0.301 0.225 0.435 0.147 0.107 0.084

0.097 0.096 0.093 0.094 0.065 0.100 0.094 0.093

3 5 1.0 0.094 0.082 0.079 0.074 0.057 0.048 0.054 0.090

0.043 0.046 0.048 0.046 0.053 0.043 0.047 0.049

0.9(A) 0.580 0.467 0.392 0.339 0.278 0.204 0.157 0.113

0.126 0.127 0.122 0.127 0.085 0.122 0.124 0.124

2 0 0 1.0 0.054 0.067 0.078 0.083 0.055 0.067 0.076 0.121

0.076 0.077 0.078 0.081 0.068 0.074 0.077 0.084

0.9(A) 0.969 0.887 0.747 0.613 0.559 0.263 0.151 0.072

0.064 0.068 0.064 0.065 0.070 0.069 0.071 0.064

0 5 1.0 0.050 0.053 0.059 0.063 0.046 0.046 0.050 0.077

0.045 0.046 0.046 0.048 0.038 0.044 0.047 0.048

0.9(A) 0.886 0.792 0.703 0.609 0.435 0.360 0.285 0.195

0.224 0.220 0.223 0.211 0.133 0.222 0.213 0.215

3 0 1.0 0.089 0.072 0.068 0.066 0.047 0.043 0.045 0.065

0.039 0.039 0.039 0.043 0.048 0.039 0.041 0.042

0.9(A) 0.659 0.537 0.452 0.388 0.502 0.280 0.226 0.164

0.185 0.186 0.184 0.179 0.088 0.190 0.187 0.174

3 5 1.0 0.044 0.052 0.057 0.063 0.046 0.046 0.051 0.079

0.046 0.046 0.047 0.048 0.037 0.045 0.048 0.048

0.9(A) 0.919 0.832 0.729 0.629 0.439 0.361 0.286 0.193

0.218 0.221 0.217 0.213 0.161 0.222 0.213 0.213

68

Table 3.4 (Continued), T = 200

DGP TESTS

Distribution of Error ��0,1�

1 0 0 1.0 0.053 0.066 0.072 0.073 0.057 0.062 0.068 0.103

0.078 0.078 0.079 0.078 0.067 0.076 0.082 0.086

0.9(A) 0.999 0.995 0.987 0.972 0.936 0.794 0.630 0.275

0.286 0.291 0.273 0.284 0.322 0.308 0.282 0.278

0 5 1.0 0.141 0.103 0.086 0.078 0.057 0.047 0.049 0.074

0.053 0.054 0.057 0.054 0.060 0.052 0.056 0.058

0.9(A) 0.755 0.666 0.597 0.533 0.726 0.497 0.436 0.358

0.395 0.394 0.391 0.389 0.139 0.399 0.390 0.382

3 0 1.0 0.112 0.090 0.079 0.080 0.056 0.054 0.061 0.087

0.065 0.067 0.065 0.070 0.064 0.064 0.070 0.069

0.9(A) 0.928 0.856 0.770 0.680 0.868 0.369 0.259 0.178

0.187 0.180 0.182 0.178 0.237 0.199 0.185 0.179

3 5 1.0 0.131 0.102 0.088 0.078 0.056 0.045 0.047 0.070

0.050 0.051 0.053 0.053 0.059 0.049 0.052 0.055

0.9(A) 0.765 0.695 0.651 0.614 0.676 0.577 0.507 0.379

0.418 0.418 0.413 0.407 0.156 0.412 0.409 0.403

2 0 0 1.0 0.053 0.066 0.072 0.073 0.057 0.062 0.068 0.103

0.078 0.078 0.079 0.078 0.067 0.076 0.082 0.086

0.9(A) 0.999 0.995 0.987 0.972 0.936 0.794 0.630 0.275

0.286 0.291 0.273 0.284 0.322 0.308 0.282 0.278

0 5 1.0 0.084 0.071 0.066 0.058 0.050 0.040 0.040 0.057

0.043 0.040 0.042 0.043 0.046 0.043 0.043 0.046

0.9(A) 0.942 0.918 0.891 0.863 0.797 0.810 0.743 0.592

0.618 0.643 0.623 0.621 0.251 0.622 0.624 0.611

3 0 1.0 0.105 0.082 0.072 0.065 0.054 0.040 0.041 0.055

0.043 0.042 0.044 0.041 0.056 0.042 0.041 0.046

0.9(A) 0.927 0.872 0.809 0.766 0.887 0.576 0.446 0.303

0.314 0.318 0.302 0.311 0.294 0.357 0.322 0.304

3 5 1.0 0.069 0.064 0.062 0.056 0.050 0.043 0.041 0.061

0.044 0.042 0.043 0.045 0.041 0.045 0.045 0.049

0.9(A) 0.973 0.962 0.950 0.936 0.776 0.844 0.784 0.590

0.631 0.653 0.637 0.629 0.282 0.623 0.631 0.612

69

Table 3.5 Finite Sample Performance using Cumulative Frequencies

TESTS Distribution of Error

N(0,1) 100 1 1.0 0.036 0.047 0.059 0.065 0.042 0.049 0.053 0.100

0.041 0.043 0.045 0.047 0.038 0.045 0.047 0.051

0.9(A) 0.910 0.757 0.596 0.473 0.428 0.221 0.169 0.106

0.125 0.121 0.118 0.120 0.120 0.119 0.119 0.113

2 1.0 0.021 0.035 0.048 0.055 0.044 0.052 0.058 0.120

0.041 0.043 0.044 0.043 0.037 0.043 0.046 0.051

0.9(A) 0.728 0.508 0.347 0.280 0.240 0.119 0.100 0.073

0.082 0.090 0.087 0.093 0.087 0.084 0.087 0.081

3 1.0 0.011 0.024 0.037 0.049 0.050 0.057 0.065 0.133

0.039 0.046 0.046 0.047 0.037 0.044 0.047 0.051

0.9(A) 0.479 0.289 0.206 0.164 0.136 0.084 0.075 0.063

0.075 0.074 0.072 0.072 0.073 0.076 0.075 0.068

200 1 1.0 0.042 0.051 0.051 0.055 0.042 0.042 0.046 0.069

0.042 0.048 0.046 0.046 0.039 0.043 0.048 0.050

0.9(A) 0.999 0.994 0.981 0.953 0.905 0.732 0.600 0.350

0.406 0.391 0.401 0.399 0.412 0.402 0.386 0.380

2 1.0 0.029 0.040 0.042 0.050 0.040 0.042 0.045 0.073

0.041 0.042 0.049 0.048 0.040 0.046 0.047 0.048

0.9(A) 0.995 0.972 0.927 0.853 0.784 0.513 0.386 0.210

0.241 0.238 0.232 0.224 0.231 0.226 0.231 0.226

3 1.0 0.019 0.031 0.037 0.044 0.044 0.043 0.045 0.075

0.043 0.045 0.046 0.049 0.041 0.045 0.047 0.047

0.9(A) 0.982 0.917 0.815 0.704 0.628 0.350 0.246 0.151

0.164 0.164 0.159 0.151 0.149 0.155 0.157 0.156

70

CONCLUSION

In this dissertation, we have tried to find ways to improve the power of unit root tests as

well as deal with unknown structural breaks. Researchers dealing with empirical data usually

have difficulties distinguish the time series they are dealing with is best characterized by a unit

root or a stationary process due to the low power problem of current unit root tests. As such, the

seeking of more powerful unit root tests is not a trivial concern.

In the first essay, we have suggested new unit root tests that are more powerful in the

presence of non-normal errors. Although usual linear ordinary least square estimation is still the

most efficient estimation under certain assumptions that do not include the normality

assumption, this does not necessarily mean that the non-normal information in the error term is

useless. The information of non-normal errors is utilized in a linearized testing regression. The

suggested tests have the same property as the tests based on the generalized methods of moments

where nonlinear moment conditions of the residuals are employed to capture unknown forms of

non-normal distributions. To do this, we adopt a two-step procedure following the RALS

method of Im and Schmidt (2008), which can make use of nonlinear moment conditions driven

by non-normal errors. The idea is simple and intuitive. If the errors follows a non-normal

distribution, the higher moments of the residual will contain information on the nature of the

non-normal errors. If we can utilize this information, we can potentially obtain more powerful

unit root tests. One advantage of RALS-LM test is that it is quite general. The knowledge of the

non-normal errors is not required, and the nonlinear moment conditions associated with the non-

71

normal distribution of the error term are generated automatically from the regression. Then these

nonlinear moment terms are augmented to the testing regression to reduce the variance of the

residual in the original regression. Since these nonlinear condition moments terms are

orthogonal to the regressors in the original regression, the estimation of the coefficients is not

changed. However, the smaller variance in the new residuals leads to a smaller confidence

interval of the coefficients and thus makes the estimator more efficient. Another nontrivial

advantage of the proposed tests is that the testing procedure is linear although we utilize the

information embedded in the nonlinear moments of the error term. Lee, Meng and Lee (2011)

found by Monte Carlo simulation that popular nonlinear unit root tests lost power significantly

when the error term follows a non-normal distribution. However, this is not a problem in the

linear models due to central limit theorem. Last but not the least important, the RALS-LM unit

root tests inherited the excellent property from LM tests that they are invariant to the nuisance

parameter denoting the location of level-break. This feature is especially important for

practitioners when multiple level breaks are found. One set of critical value is good for various

number and/or various locations of level-breaks. One caveat of RALS-LM test is that it is less

powerful than usual LM test when the error term does follow a normal distribution. But the

power loss is marginal and can be ignored without causing too much concern. Another caveat of

the RALS-LM unit root tests inherits from LM unit root tests. While DF type unit root tests

become more powerful as the initial value gets large, LM and DF-GLS type tests lost power

completely as the initial gets large. Unreported Monte Carlo simulation results show this

property and provide a warning to uses of RALS_LM tests. As such, we believe that it will be

desirable to consider a strategy to exploit the desirable properties of both RALS-LM tests and

72

RALS-DF tests. At minimum, a strategy of a union of rejection decision rules can be

considered. Such as strategy was advocated by Harvey et al. (2009) and Harvey and Leybourne

(2005) when choosing between DF-GLS and DF tests. This topic remains for future research.

Moreover, it is also interesting to extend the RALS-LM test to panel unit root tests.

The second essay tests the Prebisch-Singer hypothesis by re-examining the paths of

relative primary commodity prices that are known to exhibit multiple trend breaks. To address

the issue more properly, we improved the RALS-LM unit root tests proposed in first essay by

allowing for multiple trend breaks using a simple transformation to solve the nuisance parameter

problem. The suggested new tests have good size property and become more powerful when the

error terms follows a non-normal distribution. When applying the RALS-LM unit root tests with

multiple trend breaks to the newly extended Grilli and Yang index of 24 commodity series from

1900 to 2007, we found clear evidence of non-normal errors. We find more price series are

characterized as trend stationary with one or two trend breaks, that 21 out of the 24 commodity

prices are found to be stationary around a broken trend, implying that shocks to these

commodities tend to be transitory. However, we find less commodity price series display

significant negative trend more than 50% of the examined time period, yet, more price series

display a significant positive trend or no significant trend. Our findings provide little evidence to

support the Prebisch-Singer hypothesis. Our finding provide significant policy implications to

the policy makers, especially those in the developing countries. For those stationary

commodities, fiscal policies and/or monetary policies aimed to impact the price of these

commodities will take effect only for a short time period, in the long run, the price will go back

to it is long run mean or trend. For those countries which produce commodities have a

73

significant decline in the relative price, it is beneficial to switch to the production and export of

commodities with a significant positive trend or manufactured goods, and import these

commodities with significant negative trend. These countries will benefit from these policies in

the long run.

The third essay focus on the modeling of unknown forms of structural breaks. This essay

extends the Fourier Lagrange Multiplier (FLM) unit root tests of Enders and Lee (2012a) by

using the RALS estimation procedure of Im and Schmidt (2008). While the FLM type of tests

can be used to control for smooth structural breaks of an unknown functional form, the RALS

procedure can utilize additional higher-moment information contained in non-normal errors. For

these new tests, knowledge of the underlying type of non-normal distribution of the error term or

the precise functional form of the structure breaks is not required. By using parsimonious

number of parameters to control for possible breaks in the deterministic term and use the

additional information lied in nonlinear moments of the error term, the power of RALS-FLM test

increase dramatically when the error term follows a non-normal distribution.

74

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Becker, R., Enders, W., and Hurn, S. (2004). “A General Test for Time Dependence in

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